physics of gravitational wave detection: resonant … · physics of gravitational wave detection:...

PHYSICS OF GRAVITATIONAL WAVE DETECTION: RESONANT AND

INTERFEROMETRIC DETECTORS

Peter R. Saulson* Department of Physics

Syracuse University, Syracuse, NY 13244-l 130

ABSTRACT

I review the physics of ground-based gravitational wave detectors, and summarize the history of their development and use. Special attention is paid to the historical roots of today’s detectors.

*Supported by the National Science Foundation, under grant PHY-9602157

@ 1998 Peter R. Saulson.

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1.2 The Effect of Gravitational Waves on Test Bodies

The statement that the gravitational wave amplitude is the metric perturbation tensor h is probably hard to visualize without considering some examples. Imagine a plane in space in which a square grid has been marked out by a set of infinitesimal test masses (so that their mutual gravitational interaction can be considered negligible compared to their response to the gravitational wave). This is a prescription for embodying a section of the transverse traceless coordinate system mentioned earlier, marking out coordinates by masses that are freely-falling (i.e., that feel no non-gravitational forces).

Now imagine that a gravitational wave is incident on the set of masses, along a direction normal to the plane. Take this direction to be the z axis, and the masses to be arranged along the z and y axes. Then, if the wave has the polarization called h+, it will cause equal and opposite shifts in the formerly equal 5 and y separations between neighboring masses in the grid. That is, for one polarity of the wave, the separations of the masses along the z direction will decrease, while simultaneously the separations along the y direction will increase. When the wave oscillates to opposite polarity, the opposite effect occurs.

If, instead, a wave of polarization h, is incident on the set of test masses, then there will be (to first order in the wave amplitude) no changes in the distances between any mass and its nearest neighbors along the z and y directions. However, h, is responsible for a similar pattern of distance changes between a mass and its next-nearest neighbors along the diagonals of the grid.

There are several other aspects of the gravitational wave’s deformation of the test system that are worth pondering. Firstly, the effect on any pair of neighbors in a given direction is identical to that on any other pair. The same fractional change occurs between other pairs oriented along the same direction, no matter how large their separation. This means that a larger absolute change in separation occurs, the larger is the original separation between two test masses. This property, which we can call “tidal” because of its similarity to the effect of ordinary gravitational tides, is exploited in the design of interferometric detectors of gravitational waves.

Another aspect of this pattern that is worthy of note is that the distortion is uni- form throughout the coordinate grid. This means that any one of the test masses can be considered to be at rest, with the others moving in relation to it. In other words, a gravitational wave does not cause any absolute acceleration, only relative accelerations between masses. This, too, is fully consistent with other aspects of gravitation

Fig. 1. An array of free test masses. The open squares show the positions of the masses before the arrival of the gravitational wave. The filled squares show the positions of the masses during the passage of a gravitational wave of the plus polarization.

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First, consider light in the arm along the 2 axis. The interval between two neighboring space-time events linked by the light beam is given by

ds2 = 0 = g,,vdxpdxy

= (9y + h,,) dx”dx” (1) = -c2dt2 + (1-t hll(27rjt - kz)) dx*.

This says that the effect of the gravitational wave is to modulate the square of the distance between two neighboring points of fixed coordinate separation dx (as marked, in this gauge, by freely-falling test particles) by a fractional amount hi,.

We can evaluate the light travel time from the beam splitter to the end of the x arm by integrating the square root of Eq. (1)

where, because we will only encounter situations in which h < 1, we’ve used the binomial expansion of the square root, and dropped the utterly negligible terms with more than one power of h. We can write a similar equation for the return trip

I i7t ,& = -- ; lo (I+ $(2ajt - kz)) dx. (3) Toout

The total round trip time is thus

2L r,t = - + -

c ;, AL htl(arjt - kz)dx - ; Lo hll(2Rjt - h)dx. (4)

The integrals are to be evaluated by expressing the arguments as a function just of the position of a particular wavefront (the one that left the beam-splitter at t = 0) as it propagates through the apparatus. That is, we should make the substitution t = x/c for the outbound leg, and t = (2L - X)/C for the return leg. Corrections to these relations due to the effect of the gravitational wave itself are negligible.

A similar expression can be written for the light that travels through the y arm. The only differences are that it will depend on hsa instead of hit and will involve a different substitution fort.

If 277 jgurrrt < 1, then we can treat the metric perturbation as approximately constant during the time any given flash is present in the apparatus. There will be equal and opposite perturbations to the light travel time in the two arms. The total travel time difference will therefore be

2L b-(t) = h(t); = h(t)rrto,

where we have defined rrta G 2Lf c.

If we imagine replacing the flashing lamp with a laser that emits a coherent beam of light, we can express the travel time difference as a phase shift by comparing the travel time difference to the (reduced) period of oscillation of the light, or

Another way to say this is that the phase shift between the light that traveled in the two arms is equal to a fraction h of the total phase a light beam accumulates as it traverses the apparatus. This immediately says that the longer the optical path in the apparatus, the larger will be the phase shift due to the gravitational wave.

Thus, this gedanken experiment has demonstrated that gravitational waves do indeed have physical reality, since they can (at least in principle) be measured. Further- more, it suggests a straightforward interpretation of the dimensionless metric perturbation h. The gravitational wave amplitude gives the fractional change in the difference in light travel times along two perpendicular paths whose endpoints are marked by freely-falling test masses.

1.4 Another Way to Picture the Effect of a Gravitational Wave on

Test Bodies

In standard laboratory practice, it is not customary to define coordinates by the world- lines of freely-falling test masses. Instead, rigid rulers usually are used to do the job. The forces that make a rigid ruler rigid are something of a foreign concept in relativity, appearing ugly and awkward after the gravitational force has been made to disappear by expressing it as the curvature of space-time. On the other hand, non-gravitational forces are not only a fact of nature, but part of the familiar world of the laboratory. For many purposes, it is convenient to retreat from a purely relativistic picture and instead use a Newtonian picture in which gravity is treated as force on the same level as other forces.

What we are seeking is not a different theory of gravitational waves, but a trans- lation of the theory discussed in the previous section into more familiar language. So let us reconsider the same gedanken experiment as before, but imagine that we have augmented the equipment with a rigid ruler along each axis. We saw that when a gravitational wave passed through our set of test masses, the amount of time it took for light to travel from the vertex mass to the end mass and back was made to vary. How can we

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Before we rush to plug in a distance R of a few meters, as Hertz was able to do for his experiment, we need to remember that wave phenomena are only distinguishable from near-field effects in the “wave zone,” that is, at distances from the source comparable to or larger than one wavelength. With w,.,~ = 27r x 1 kHz, we have A = 300 km! The receiver for our Hertzian experiment must be at least that far away from the transmitter. Hertz’s electromagnetic experiments involved waves of six meters down to 60 cm in length, so the distance across the lab was fine for him.

At a distance of one wavelength, our laboratory generator gives gravitational waves of amplitude

blab = 9 x 10-39. (10)

This is pretty small Even creating such a strong source as this may not be practicable. Consider the

stress in the connecting rod of the dumbbell. It must supply the centripetal force necessary for the masses to move in a circle. If the rod were made of good steel, it would need a cross-sectional area substantially greater than that of a one ton sphere in order not to fail under the stresses in a device with the parameters we have assumed. So we’d have to reduce the rotation frequency to keep the generator from flying apart, with a consequent reduction in the transmitted wave amplitude.

2.2 Astrophysical Sources of Gravitational Waves

Even if a gravitational version of the Hertz experiment is not feasible, all is not lost for the detection of gravitational waves. The best reason for optimism that detectable levels of gravitational radiation exist comes from the presence in the universe of objects with truly remarkable values of i’. These systems are so extreme that even though their distances from our detectors are quite large, they still generate gravitational waves with amplitudes that exceed by almost 20 orders of magnitude the signal strengths from laboratory generators of the type described above.

It would be beyond the scope of this review to describe in detail all of the many astronomical objects that might be important sources of gravitational waves. Readers are urged to consult the article by Finn in these proceedings for further information on the variety of possible sources. But for the sake of a self-contained treatment, we show here how to estimate the magnitude of the strongest gravitational waves arriving at the Earth.

For the case of a binary star, there is an elegant way (due to Kafka3) of writing

the amplitude of the quasi-sinusoidal gravitational wave strain. We can massage the quadrupole formula into a manifestly dimensionless form by recognizing that the mass dependence can be rewritten as a proportionality to the product of the Schwarzschild radii R, = 2GM/c2 of the stars. The frequency dependence and all of the stray factors remaining collect nicely as the separation r of the two stars. The gravitational wave amplitude is

h,, = RslRs2/~R. (11)

If the binary consists of two neutron stars, then the Schwarzschild radii are both about 4 km. Astronomers estimate that within a sphere of radius 200 Mpc, roughly one of these sytems will coalesce each year. When the stars have a separation of ten diameters (or around 200 km), then the signal we would receive from that distance will have an amplitude of almost 10-23. The stars can probably approach closer still before the system is destroyed.

A glance at this expression shows why a neutron star binary is a good choice as a strong source of gravitational waves. The substantial masses of the two stars make the numerator large. The fact that they are compact objects means that their separation r can be quite small. We could always wish that the distance R to the nearest example of such a system were smaller, but even so our estimated signal strength, while small in absolute terms, is certain dramatically larger than we were able to produce in our model laboratory generator.

Perhaps the only sort of astronomical system we can imagine that might generate stronger gravitational waves would be a binary system consisting of two black holes. Although it may be hazardous to treat such dramatically relativistic objects with the quasi-Newtonian physics used to derive Eq. (1 l), it will probably still give a good order of magnitude estimate. The possible advantages of black holes as sources of gravitational waves are twofold. Firstly, it is possible that the masses of black holes may be substantially in excess of the 1.4 M0 typical of neutron stars. Secondly, black holes can approach to a separation T as close as their Schwarzchild radius R, without disruption; instead the two will coalesce into a single larger black hole. Thus we guess that the gravitational signal from a black hole coalescence could be as large as

hbh - RJR. (12)

For a pair of 10 M. black holes at 200 Mpc, this expression would indicate a signal of h-5x 10-21.

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then becomes that of a simple harmonic oscillator, with the driving term given by the effective force from the gravitational wave [our Eq. (7)].

Weber next shows how an extended elastic body behaves in such a way that each of its normal modes of vibration can be studied independently. (The gravest mode of a cylinder has a large quadrupole moment, and is the one that is usually used for detection.) He focuses attention on the use of a piezoelectric crystal as the detecting body, partly because he hopes that the electric field will make it a detector with effective size larger than half an acoustic wavelength, but also in large measure because the electric fields generated by the gravitational-wave-induced stress will give an integrated voltage between its ends that may be “large enough to be observed with a low-noise radio receiver.” Weber calculates the amount of mechanical power that a sinusoidal gravitational wave can dissipate in the resonant detector as a function of frequency, then invokes the standard electrical network theorems to show what fraction of this power can be transferred to the input impedance of an amplifier.

A simple discussion of sensitivity follows. Weber first remarks that “in microwave spectroscopy it has been found that all spurious effects other than random fluctuations can be recognized.” Then Weber states that the excitation of the detector must exceed the noise power associated with its thermal excitation.

Finally, Weber discusses possible practical experimental arrangements. In most of the discussion the devices are supposed to be made of large blocks of piezoelectric material. But in a footnote Weber states that the experimental work he is carrying out with David Zipoy and Robert L. Forward will probably make use of a large block of metal instead. (This is justified on the grounds that a half-wavelength at the 1 kHz frequency being contemplated is already large; thus the piezoelectric length-enhancement effect may not be necessary, and in any case such a large block of piezoelectric material “may not be obtainable as a single crystal”.)

Two experimental strategies are foreseen: use of a single detector with examination of its output for a diurnal cycle associated with the scanning of its sensitivity pattern across the sky, and the cross-correlation of a pair of detectors so that external influences (presumably gravitational waves) can be distinguished from “internal fluctuations.” He notes the necessity of preventing the excitation of the detector by “earth vibrations,” and discusses an “ingenious” idea of Zipoy’s for what is now called active vibration isolation.

Weber’s very concise discussion is remarkable for the prescience with which it for- shadowed not only his own work, but that of so many others. It also marks a watershed

in the history of general relativity. In a single blow, Weber wrested consideration of gravitational waves from theorists concerned about issues such as exact solutions, and appropriated the subject instead for experimentalists trained in issues of radio engineering. The boldness and brilliance of this move are remarkable.

3.2 The Logic of Weber’s Idea

Weber sweeps quickly over a variety of issues that are worthy of more leisurely consideration. We’ll give an overview of the important issues in this section, then devote the rest of this review to discussing their implications.

The detector Weber outlined can be divided into several subsystems: a set of test masses that respond to the gravitational wave, a transduction system that converts this mechanical response to a convenient electrical signal, a low-noise preamplifier, and a post-amplification averaging and recording mechanism. Notwithstanding the cleverness of Weber’s original version, many variations on his basic scheme are possible, and indeed are responsible for much of the progress since he first announced the results of gravitational wave observations in 1969.’

Let’s see how to analyze the original Weber design into these canonical subsystems. Weber explicitly pointed out how one could construct an analog of a pair of lumped test masses by monitoring an internal mode of vibration of an extended block of elastic material. In the version where this block is made of piezoelectric material, the same material serves both as test masses and as transducer from mechanical to electrical signal form. In the version in Weber’s footnote (the one he actually built) a large aluminum cylinder serves as the set of test masses; piezoelectric strain gauges glued about the girth of the cylinder perform the transduction. The pre-amplifier is Weber’s low- noise radio receiver. No averaging filter is shown in Weber’s diagrams, but is implicit in his discussion.

Perhaps the most interesting choice that Weber made was to connect his test masses in a resonant system. It appears that Weber, at least in 1961, thought this was a necessity. In a footnote, he cites previous work by Piranig in which the latter considered “measurement of the Riemann tensor by comparing accelerations of free test particles,” but Weber continues, “The results of this chapter indicate that interacting particles must be used, in practice.” In fact, it is not required either in principle or in practice, but it is interesting to consider why Weber may have thought so then, and what advantages still accrue to the use of resonant masses.

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such a system has a low post-detection bandwidth (usually shortened simply to “bandwidth.“) The averaging washes out any details of the waveform h(t) on time scales short compared to 74. What one gains in signal-to-noise ratio, one gives up in temporal resolution. Whether this is a price one ought to be willing to pay or not depends on the stakes: if it is absolutely necessary even to detect the signal, averaging with a matched filter is certainly worthwhile. If the signal could be detected anyway, averaging simply throws away information, and should be avoided. In the high signal-to-noise case, the resonance does not help, but neither does it hurt much-a simple filtering operation could remove the resonant signature and allow reconstruction of the original signal waveform.

(N.B.: As we will show below, the actual choice of matched filter for a resonant detector is more subtle than that just described. Instead of rd, a shorter averaging time is almost always the optimum choice. Nevertheless, the qualitative thrust of the argument given in the previous paragraph still applies.)

3.4 Free-Mass Detectors as an Alternative

Given the trade-off between sensitivity and bandwidth that resonant systems tempt one to make, it is worth exploring whether there are other entirely non-resonant detection schemes that can achieve high sensitivity to gravitational waves without sacrificing signal bandwidth. In fact, such free-mass detectors have been developed by a variety of workers, including the same Robert Forward who worked with Weber on the original resonant detector.‘0x” The essential advantage of free-mass detectors comes from the fact that the farther apart their test masses are placed, the larger is the relative displacement between them caused by a given gravitational wave amplitude h(t). (This scaling relation holds true up to the point that the light travel time between the masses becomes comparable to the period of the wave; that is when separation of the masses becomes comparable to the wavelength of the wave.) But the resonance in a resonant detector comes roughly when the sound travel time across the bar matches the period of the wave. That is to say, resonant detectors reach their optimum sensitivity when the separation of the test masses is of order of the acoustic wavelength at the gravitational wave frequency. Since the speed of sound in materials is of order 10d5 of the speed of light, a free-mass detector at its optimum length can have an advantage in signal size of lo5 over a resonant-mass detector at its optimum length.

Another advantage is that no resonance is used to boost the signal. Thus, in principle

a free-mass detector can have a completely white frequency response. This ideal can not be completely achieved in practice, since some of the noise sources discussed below have strong frequency dependences of their own. Still, it is possible to achieve useful bandwidths measured in decades rather than in fractions of an octave.

This signal size advantage would be a hollow one if there were no sensitive way to measure the relative displacement of test masses separated by many kilometers. For- tunately, there are such ways. As we saw above, the travel time of electromagnetic signals between the test masses can be measured with great precision. Interferometry using visible or near-infrared light to measure the separation of free masses has become a well-developed technology that now is completely competitive with the best resonant-mass detectors, and which is about to undergo a great leap in sensitivity as new instruments of multi-kilometer scale come on line in the next couple of years. Radio ranging between interplanetary space probes separated by many millions of kilometers has been used for some time; optical interferometers in solar orbit, with million kilometer baselines, are now being planned.

The conceptually simpler free-mass detectors are in practice substantially more complicated devices; the freedom of the test masses must be tamed by servo systems to keep them operating properly. This is in part what is responsible for the time lag in their development, even though they were conceived not much later than resonant-mass detectors. In the remainder of the review, we will discuss both styles of gravitational wave detector.

4 Noise Sources

In this section, we will focus our attention on understanding the most fundamental noise sources with which the practice of gravitational wave detection has to contend. Perhaps not surprisingly, the list will seem to have little to do with general relativity or with gravitational waves, as such. The chief concerns of gravitational wave detector designers are those that would confront anyone attempting to measure the effect of a very weak force on a mechanical system: Brownian motion (also known as thermal noise), and noise from the readout system (both in its direct influence on the output of the system and through its “back-reaction” on the mechanical front end). A ubiquitous but non-fundamental noise source, seismically-induced vibration, is treated as well.

It is pedagogically simpler to introduce the topics first in the context of interferometers. Then, we will describe how similar considerations apply to resonant-mass

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related concept called the admittance Y is defined by

Y(w) E z-‘(w) = $)3i”.

With these preliminaries, Callen’s Fluctuation-Dissipation Theorem can be suc- cinctly stated. The thermodynamic fluctuations analogous to Brownian motion have a magnitude given by the application at the point of interest of a random force with a power spectrum

SF(w) = 4kBTRe(Z). (14)

The strength of the applied force power spectrum is proportional to the dissipative (real) part of the impedance; hence the name “fluctation-dissipation” theorem. Note that this expression has the same form as the more familiar power spectrum for the Johnson noise voltage, S”(w) = 4kBTR, where the resistance R is the real part of the electrical impedance. The similarity is not accidental, but is only one example of the many phenomena unified by the theorem.

An alternative form of the theorem, more useful in some situations, directly gives the displacement fluctuation power spectrum instead of the equivalent applied noise force. It states

&(w) = Y&(Y). (15)

Again, the power spectrum scales with the amount of dissipation in the system. Clearly, this description of fluctuation phenomena is richer than the Equipartition

Theorem, since here we have expressions for the entire power spectrum of the fluctuations, not just their rms amplitude. But are the two descriptions even consistent? The rms fluctuation, for example, the expression in Eq. (13) has no dependence on the magnitude of the dissipation. But Eq. (15) shows that the fluctuation power spectrum is proportional at each frequency to the amount of dissipation at that frequency. How can both be true? An oscillator with low dissipation shows a very pronounced peak in its response at the resonance frequency, while one with larger dissipation exhibits a less dramatic peak. So, although the driving noise force is smaller when the dissipation is smaller, the response on resonance is greater. The two effects precisely cancel, as can be verified by direct integration, thus guaranteeing that the integral of the power spectrum Eq. (15) is equal to what one would predict from the Equipartition Theorem.

These two faces of thermal noise, rms magnitude, and power spectrum, are each important in the appropriate context. In a broad-band gravitational wave detector, such as one using an interferometer, the power spectrum carries the most valuable information.

This insight is embodied in the universal choice to suspend the test masses as pendulums. Pendulums are chosen because they are the best way known to create a low frequency oscillator with very low dissipation. Heuristically, most of the restoring force in a pendulum comes from the tension in its wires (due in turn to the gravitational force on the mass); this process has no dissipation associated with it. The only unavoidable dissipation is that associated with the flexure of the wires, but in a properly designed pendulum the fraction of restoring force associated with flexure is small. Hence, the internal friction in the wires is “diluted” by a large factor (perhaps of order 103).

Similarly, one wants to minimize the thermal noise associated with internal vibrations of the test masses. This can be achieved only by making the masses out of a material with very low dissipation. Fortuitously, fused silica has very low mechanical dissipation at acoustic frequencies at room temperature.

A standard design rule in those devices is to attempt to place all resonances (such as those associated with the pendulum suspension of the test masses or those involving internal vibrations of the test masses themselves) outside of the frequency band in which signals will lie. When this is done, only the off-resonance amplitude of the power spectrum is important. The off-resonance transfer function of an oscillator to a given force is controlled by the compliance of the resonator in the low frequency limit, and by the inertia of the oscillator above resonance. If the dissipation that sets the driving force can be made low, so can the power spectrum of thermal noise at the frequencies of interest.

4.2 Readout Noise and the Quantum Limit

All experiments need readout and recording systems to register the effects for which we are searching. If the effect is large enough, then these functions can be carried out essentially perfectly. But in the case of the tiny mechanical effects we expect from gravitational waves, even to make the mechanical system’s response large enough to record requires very carefully designed readout systems. It is not possible in all cases to ensure that the noise in the readout system is small compared to the mechanical noise in the test masses.

Readout noise has two faces, either one of which may dominate depending on the circumstances. The most familiar is additive noise that competes with a fair copy of the mechanical signal in the output of the measuring system. But measurement systems also unavoidably add mechanical noise to the front end; this “back reaction” noise

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which half of the maximum possible power exits the output port. At this point, the change in output power is maximized for a given change in path length difference. If we want to observe a very small change in arm length difference, then we must be able to recognize a very small change in the output power of the interferometer. In other words, the readout precision of an interferometer is limited by the precision with which we can measure optical power.

The fundamental limit to this power measurement is the so-called “shot noise” in the light. We can model the light flux at the photodetector as a set of discrete photons whose arrival times at the photodetector are statistically independent, although with a deterministic mean rate fi. Whenever we count a number of discrete independent events characterized by a mean number N per counting interval, the set of outcomes is characterized by a probability distribution p(N) called the Poisson distribution,

p(N) zz 7. (17)

(This is also colloquially referred to as “counting statistics.“) When i%’ > 1, the Poisson distribution can be approximated by a Gaussian distribution with a standard deviation u equal to fi.

We are trying to determine the rate of arrival of photons 6 (with units of set-I), by making a set of measurements each lasting r seconds. The mean number of photons in each measurement interval is &’ = fir. The Poisson fluctations of the measurement process mean that the fractional precision of a single measurement of the photon arrival rate (or, equivalently, of the power) is given by

“I”-m N no =&. (18)

This says that if we were to try to estimate ii from measurements for which fir N 1, then the fluctuations from instance to instance will be of order unity. If fir is very large, then the fractional fluctuations are small.

Let’s carry through the calculation for the power fluctuations, and thence to the noise in measurements of h. Each photon carries an energy of Aw = 2rfic/X. If there is a power Pout at the output of the interferometer, then the mean photon flux at the output will be

n = LP&. 27riic

At the half-power operating point,

(19)

(20)

We can also consider this to be the sensitivity to the test mass position diference bL,

since the interferometer is equally sensitive (with opposite signs) to shifts in the length of either arm.

Now consider the fluctuations in the mean output power Pat = Pin/S, averaged over an interval 7. The mean number of photons per interval is E = (X/4~hc)P,,7.

Thus we expect a fractional photon number fluctuation of us/N = dw. Since we are using the output power as a monitor of test mass position difference, we would interpret such statistical power fluctuations as equivalent to position difference fluctuations of a magnitude given by the fractional photon number fluctuation divided by the fractional output power change per unit position difference, or

Recall that we can describe the effect of a gravitational wave of amplitude h as equivalent to a fractional length change in one ~~III of AL/L = h/2, along with an equal and opposite change in the orthogonal arm. The net change in test mass position difference is 6L = Lh, so if we interpret brightness fluctuations in terms of the equivalent gravitational wave noise ah, we have gh = O&L/L, or

(22)

There is no preferred frequency scale to this noise; the arrival of each photon is independent of the arrival of each of the others. Note also that the error in h scales

inversely with the square root of the integration time. These facts can be summarized by rewriting Eq. (22) as the statement that the photon shot noise in h is described by a white amplitude spectral density of magnitude

(23)

4.2.3 Radiation Pressure Noise in an Interferometer

A hint at where quantum mechanics might have some deep relevance comes when we consider how shot noise scales with the optical power used in the interferometer. As shown in Eq. 23 above, the shot noise readout precision improves as the square root of the optical power. Taken at face value, this would suggest that we could achieve arbitrarily good measurement precision, so long as we were able to use an arbitrarily powerful laser to illuminate the interferometer.

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There was a moment when some physicists believed, on seemingly sound physical grounds, that this picture of how photons interact with a beam splitter was so flawed that interferometers could perhaps evade the Uncertainty Principle.” The argument can be made based on quotation from quantum mechanical Scripture, Dirac’s The Princi- ples of Quantum Mechanics. ig There one can read that photons in an interferometer travel down both arms simultaneously; furthermore, it is written that interference can only take place between a photon and itself, so the very existence of interference in a quantum mechanical world is proof of this picture. If this were taken as absolute and literal truth, then it would appear to rule out any differential radiation pressure at all, since the number of photons, and hence the recoil forces, would be identical in the two arms. Without the resulting differential recoil of the test masses, there is no quantum limit. Gravitational waves could in principle be measured with arbitrary precision. Some physicists defended this as gospel, despite the fact that the argument appeared to use quantum mechanical reasoning to disprove quantum mechanics.

The stubbornly naive were untroubled by this argument, and expected the Uncer- tainty Principle to hold. Some physicists read a few pages further in Dirac’s book, to the passage explaining that allowing the possibility of energy measurements, say by observation of recoil of the mirrors, causes collapse of the wave function in such a way that photons end up either in one arm or the other. (Dirac’s first discussion refers to an interferometer with rigidly fixed mirrors.) The learned were saved from error by the work of Caves,” who invoked the concept of vacuum fluctuations to explain the quantum mechanical behavior of photons at a beam splitter. A vacuum electromagnetic field with zero-point fluctuations enters the interferometer through the output port; its superposition with the field from the laser causes the light to behave in the way expected from semi-classical reasoning.

4.3 Seismic Noise

We have neglected to consider above another source of noise in gravitational wave detectors that is so common and important as to be essentially ubiquitous. This is what is commonly called seismic noise, the continual shaking of the terrestrial environment due to a variety of contingent causes, ranging from small earthquakes to ocean waves driven by large weather systems to automobiles striking potholes in poorly paved streets. Such a complex phenomenon can have no simple explanation from basic physics, yet dealing with it forms a substantial part of the challenge to designers of gravitational wave de-

tectors. (Only moving the whole detector into space suffices to remove it entirely from consideration.)

At a reasonably quiet location, the spectrum of seismic noise from 1 Hz to several hundred Hz can be approximated as

x(f) = 10-7cm/JHz, from 1 to 10 Hz 10-7cm/&(10Hz/f)“, for f > 10 Hz.

(31)

The magnitude of this mechanical noise background is distressingly large. The rms amplitude of the noise over this interval is of order 1 pm. The good news is that the spectrum falls with increasing frequency f. But even so, throughout the range of frequencies of interest to gravitational wave detectors, it involves motions many orders of magnitude larger than would be driven by any conceivable incident gravitational wave. There is no possibility of success unless the effects of seismic noise can be strongly attenuated.

It is straightforward to see the way in which seismic noise mimics a gravitational wave signal in an interferometer. As long as the separation between the mirrors is not very small, then the seismic inputs to each mirror are effectively independent; the difference in arm lengths is driven by the quadrature sum of the noise at all mirrors. The situation is a bit more subtle for a resonant mass detector. If it is suspended at its midpoint, it would appear that its internal modes should not be excited by any motion of the suspension point. However, this argument assumes perfect symmetry of the resonator about the suspension. The approximate symmetry of real systems may give several orders of magnitude of effective isolation, but the seismic spectrum is so large that additional isolation is always required.

Fortunately, the design of seismic isolators is a well-developed art. One can construct mechanical multi-pole low-pass filters that provide outstanding attenuation at frequencies well above those of the filter poles. The art of doing so was introduced to the field of gravitational wave detection by the founder, Weber.*’

4.3.1 A Simple Two-Pole Isolator

The essence of vibration isolation can be understood using only ideas from freshman physics. Imagine that the object to be isolated has mass m. Assume that it is a rigid body, and that we are only interested in its motion 2, in a single direction. Then we can treat the object as a point mass. If it rests on the ground, it shares the ground’s motion xg, so x, = x9. To isolate the mass, replace the rigid connection to the ground

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4.4 Noise in Resonant-Mass Detectors

Key features The essential complication in understanding resonant-mass detectors (as compared to interferometric detectors) is that the degree of freedom of interest is that of a simple harmonic oscillator (or a collection of them, as we’ll see in the next section). So in addition to any intrinsic frequency dependence in the noise, there is a deliberately constructed resonant transfer function in the detector itself. As we saw earlier in this review, the resonance was introduced as part of a strategy for overcoming wide-band noise in the amplifier.

The use of this strategy involves different heuristic concepts than are appropriate for interferometers. In particular, optimizing the sensitivity of a resonant detector to short bursts almost always involves choosing to average the output over times that are long compared with the length of the burst itself. Then, the measurable quantity is no longer h(t), but is instead net change in the vector amplitude (magnitude and phase) of the resonator’s oscillation. This in turn can be expressed in terms of the energy that the wave would have deposited in a resonator at rest. 23 If the gravitational waveform h(t) has a Fourier transform H(f), then that excitation energy E is (for an optimal orientation between bar and wave)24

E = $ IH (fo)l’ , where M is the total mass of the bar, v, is the speed of sound in the material, L is the overall length of the bar, and fo is the resonant frequency.

The distinctive features are twofold: characterization of,all candidate events by a single number (usually its “energy” or else T E E/kc), and a signal-to-noise optimization that involves choosing the right averaging time (or bandwidth).

Resonant transducers The second generation of resonant-mass detectors replaced Weber’s piezoelectric transducer with a kind of a bridge circuit, in which the mechanical motion unbalanced the bridge by modulating the inductance or capacitance of one leg of the bridge. As with piezoelectric transducers, achieving a high level of coupling has proven difficult to achieve. A standard measure of the coupling is the Gibbons- Hawking parameter p, defined as “the proportion of elastic energy of the detector that can be extracted electrically from the transducer in one cycle.“25 In principle, the excitation of the bridge could be increased without limit, but in practice large fields usually lead to excess dissipation in the transducer even before electrical breakdown occurs. Transducers have been limited to working values of/l of around lo-‘.

A heuristic way of understanding the design problem is to think of the issue as an attempt to design a transducer that makes a reasonable electrical impedance at its output appear to the mechanical system as a mechanical resistance sufficient to supply appreciable damping to the bar. With bar masses in excess of one ton, this may seem inordinately difficult. The good values mentioned in the previous paragraph avoided this problem by making use of so-called resonant transducers, which have been adopted almost universally since the idea was proposed by Paik in 1 976.26

Paik’s design called for a smaller mechanical resonator to be attached to the main resonant mass M. The resonant frequency of the smaller resonator itself (i.e., with the larger resonator “clamped”) is chosen to match that of the main resonator. The actual coupled system then has two normal modes. If the mass ratio m/M = (Y < 1, then it is easy to show that the ratio of the amplitude of motion of the small mass, compared with that of the main resonator, is

I I $ =;. - When a gravitational wave burst interacts with such a resonant system, it will at

first mainly excite the vibration of the large bar. (The Paik resonator is a small device at one end of the bar, so the gravitational wave strain has only a small baseline for creating a stretch in its spring.) The free motion of this two-mode system then exhibits “beats,” during which the mechanical energy of the main resonator’s original motion is transferred into excitation of the small resonator. During this phase of the beat cycle, the effect of the gravitational wave has been transformed into a motion 2/& times larger than in a detector without the resonant transducer. The electro-mechanical transducer is mounted so as to measure the motion of the small mass with respect to the end of the main resonator, thus presenting this larger motion to the rest of the signal processing system.

The advantage this offers in detecting weak signals is probably obvious. The larger motion generates a comparably larger electrical output from the transducer, reducing the importance of a given level of electrical amplifier noise. Another way of seeing the advantage is to recognize the much smaller mechanical impedance required to damp the motion of the smaller mass, which means that /3 is increased by a factor of order (Y-‘. In present day designs, the mass ratio a is typically of order a few times 10m3. The value is set in an optimization that involves not only thermal noise and additive amplifier noise but the back-action noise as well.

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possible. To describe the noise, and to understand how it affects the detection process, it is valuable to consider a general black box representation of a noisy amplifier as a kind of a two-port network, just as we did for the transducer. Other than the trivial difference that this two-port is an all-electric device, there are two key differences between this kind of two-port and the model transducers discussed above. One is the fact that the amplifier has gain, i.e., it can supply more energy at its output than is supplied by its input. The other is the existence of two sources that represent the generation of noise. They are usually represented as a voltage source and a current source at the input of the two-port network; this is especially convenient, and has wide generality, but is only one of several equivalent ways of representing the noise. More on the general theory of noisy two-port networks can be found in the pioneering paper by Rothe and Dahlke.31

It is interesting to pause to inquire why two noise sources are necessary. Recall the Hehnholtz theorem, often known in specialized forms as Thtvenin’s theorem or Norton’s theorem.32 The essence of the theorem is that an arbitrary network of sources and passive components can be represented, as far as its behavior at a given port is concerned, by a single source and a single impedance. But we are dealing here with a network in which two ports are relevant. At each port, one needs an impedance and a source, or their equivalents elsewhere in the circuit. It is traditional to replace the source at the output with an equivalent noise source at the input, whose strength is smaller than the output noise by a factor of the amplifier gain.

These two noise sources play different roles in the measurement process. There is one noise source that is physically present at the input, causing an influence on the system (here the electromechanical transducer) that is attached to the amplifier input. In the jargon of gravitational wave detection, this noise is responsible for “back action,” since noise at the input of the pre-amp is thereby applied to the output end of the transducer,where it can cause a mechanical noise force at the transducer input; this is in turn attached to the resonant-mass detector proper. More on this below.

The second noise source (the one replacing the output noise source) is usually referred to as “additive noise”: it is added to the amplified signal by the time it appears at the output, without causing any physical effect on the system hooked up to the input.

Amplifier noise in resonant-mass gravitational wave detectors, and the “amplifier limit” In contrast to our discussion of detection strategies in the case where thermal noise dominates, here we discuss the case when additive amplifier noise is the only important noise source. Then, we can best search for a brief burst of gravitational

radiation by performing a cross-correlation between the system output and a template consisting of a sinusoid at the mechanical resonant frequency that is damped with the same time constant as the resonance itself. In other words, we look for responses that look like the test mass system suddenly set into resonance. The signal can arrive with any phase of course, so we need to keep track of both sine and cosine components with the bar’s damping time. A two-phase lock-in amplifier can be set up to perform exactly this form of averaging.

In this case the energy sensitivity of the detector is given byz7

where T, is the noise temperature of the amplifier, X is the ratio of the transducer output impedance to the amplifer noise impedance, p is the Gibbons-Hawking coupling parameter, and ~~ is the averaging time.

Consider the post-detection bandwidth implied by this prescription. The output of the cross-correlation described above is hardly affected if we displace the template with respect to the signal time series by one or even several cycles of oscillation. For there to be a substantial change in the value of the cross-correlation, the template must be displaced by a duration of order the damping time of the mechanical resonance. This means that, if the signal-to-noise ratio is not large, the arrival time of the impulsive gravitational wave signal will be uncertain by of order the damping time. In other words, the post-detection bandwidth Af of such a signal extraction system is narrow, of order

Af z=z l/rd. For a quality factor of 106, this bandwidth is very narrow indeed. Increased bandwidth could be achieved, of course, at the expense of the signal-to-noise ratio, by averaging the output time series for a shorter time than the bar’s damping time 7d.

Combined optimum in the presence of thermal noise and additive amplifier noise When both kinds of noise are present at substantial levels, the best strategy is neither the rapid readout appropriate to thermal noise nor the long averaging time that would be best for amplifier noise. A broad-band output filter will admit too much amplifier noise. Using the narrow-band prescription appropriate to the amplifier-dominated case filters out much thermal noise, but also most of the signal power. Obviously, the optimum in the combined noise case lies somewhere between the extremes, where the net

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From this, Giffard shows that the Uncertainty Principle requires that unless a gravitational wave signal has a minimum size, no linear gravitational wave detector will be able to register its arrival. He expressed the minimum size in terms of the energy U, that the wave would deposit in a resonant detector initially at rest. The quantum mechanical limiting sensitivity is

u, > 47&f.

As worded above, this argument can seem rather abstract. A heuristic description makes it as vivid as any of Bohr’s gedanken experiments. In a mechanical amplifier, a crucial quantum mechanical role is played by the back action from the noise source that the Helmholtz Theorem places at the amplifier input. The force noise generator at the mechanical amplifier input (caused by electrical noise acting backwards through the transducer) perturbs the delicate mechanical system, here the resonant-mass system itself. Just as in the Heisenberg microscope, any design trade-off made in an attempt to reduce the position noise ends up increasing the momentum impulse applied to the system being measured.

5 History of Resonant-Mass Detectors

We now turn from a discussion of physics per se to a review of the way in which one assembles working gravitational wave detectors in light of the physical principles governing them. We will take a quasi-historical framework for this discussion, as a pedagogically sensible way of grappling with the issues involved. In this section, we will take the chronologically-motivated choice of treating resonant-mass detectors first; then we will start from scratch the overlapping history of interferometric detectors.

By 1966, Joseph Weber had constructed a complete working detector, and by 1968 was reporting coincident observations between detectors separated by 1000 km. The detector contained versions of every essential feature in resonant-mass gravitational wave detectors today, except for the facts that it operated at room temperature and that it used non-resonant strain transducers for its readout. The story of the development of the field since then can be seen as embodying a few key accomplishments: replication of Weber’s detectors accompanied by a failure to confirm his claimed detection, clarifi- cation of the optimum way to detect gravitational wave signals in a noisy detector and of the sorts of technological developments that could lead to improved detector sensitivity, and the staged implementation of new generations of detectors embodying the

improved technology.

5.1 Weber’s Detectors as Gravitational-Wave Detection Systems

All of the important elements that make up a working gravitational wave detector are described in Weber’s 1966 Physical Review Letters. 21 Looking at the signal chain from the front end, we see first the large 15 ton aluminum cylinder whose fundamental lon- gitudinal mode at 1657 Hz interacts with any incoming gravitational wave. Around its midsection are glued the quartz transducers that give, through the piezoelectric effect, electrical signals proportional to the strain in the aluminum cylinder. Signal leads from those transducers pass through acoustic filters and through the wall of the vacuum chamber, then are connected via another acoustic filter to a superconducting inductance that serves as a “tank circuit” at the input of the low-noise preamplifier (whose noise temperature is 50 K.) The output of the pre-amp is connected to further amplification. There follows a rectifier for generating a positive-definite signal proportional to the power out of the amplifier. The end of the signal chain is a recording device, which in 1966 consisted of a pen-and-ink chart recorder.

Another aspect of the detector is the means used to prevent its being excited by influences other than gravitational waves. Isolation against mechanical influences in the form of acoustic or seismic noise is shown clearly in the diagram. Direct acoustic excitation is prevented by the placement of the key parts of the experiment inside a vacuum chamber. Transmission of vibration through the signal leads is attenuated by the acoustic filters mentioned above. The path for vibrations from the floor must pass through a pair of isolation stacks each consisting of three stages of rubber pad/steel block isolators; the aluminum bar is further isolated by a pendulum suspension consisting of a single wire sling that supports the bar about its middle. The top ends of the wire are attached to a beam that spans the space between the isolation stacks.

Further progress in reducing the sensitivity of his bar to spurious external influences enabled Weber to make his 1969 claim of “Evidence for discovery of gravitational radiation”.’ One aspect of this progress consisted of augmenting the electromagnetic shielding of his devices, after tests revealed some sensitivity. A seismometer array was also used to check for correlations between strong vibration of the ground and large detector outputs. But by far the most important system element in this regard was the construction of multiple copies of the complete detector system, and their deployment at spatially separated locations. The network in 1969 consisted of one bar of 66 cm

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epoch in the search for gravitational waves. Gibbons and Hawking, both renowned for work rather far removed from experimental physics, wrote in late 1970 the very practical “Theory of the Detection of Short Bursts of Gravitational Radiation.“” The introduction presents Weber’s results as established facts. Section II of the paper discusses possible sources, discussing with equanimity but at some length the extreme gravitational luminosities required. The rest of the paper is devoted to the theory of detecting weak gravitational wave bursts in the presence of noise, and to a clear and original set of proposals for maximizing the signal-to-noise ratio of detectors. Some of one’s surprise at the authorship of this paper may be alleviated upon reading the thanks for insights and ideas given (at two places in the text and in the Acknowledgment) to P. Aplin of Bristol University, a very original experimentalist who published little on this subject under his own name.40

Gibbons and Hawking pointed out that Weber’s own treatment of the theory of gravitational wave detectors, written before he started his observations, had been aimed at their response to steady sinusoidal signals. As such, it gives a misleading idea of the value of the high-Q resonance that characterizes Weber-style detectors. This lack was remedied by Gibbons and Hawking, who (with thanks to Aplin) point out that the low dissipation of a high-Q system means that the level of thermal noise power is low. They go on to show that the high-Q resonance is also of crucial importance in minimizing the effect of Johnson noise in the transducer, here playing the role we would generally call amplifier noise. (A weakness of this light treatment of amplifier noise is that it leaves out the back action effects that enforce the Uncertainty Principle.)

As we saw previously, Gibbons and Hawking noted that there is in fact a competition between these two benefits of low dissipation. To minimize thermal noise, one wants to integrate the output for as short a time as possible, to give the random walk of the resonator’s complex amplitude the least opportunity to mask a signal. On the other hand, the importance of transducer/amplifier noise is minimized by integrating for as long as practicable, so that the gravitational wave signal competes with as small a bandwidth of the broadband noise as possible. Gibbons and Hawking showed how to derive the optimum averaging time that minimizes the total noise from these two sources. In so doing, they noted that the averaging time depends on the dimensionless coupling parameter they called /I, the definition of which we gave above in Section 4. For Weber’s detector they give an estimate of /3 % 5 x 10e6.

Gibbons and Hawking go on to note that a large value for p would have two benefits: improving the signal-to-noise ratio by making the gravitational wave signal appear

as a larger electrical signal, while simultaneously changing the balance between thermal noise and electrical noise in the direction that causes the optimum sensitivity to be obtained with shorter integration times. In other words, a larger /3 would yield better sensitivity and high bandwidth. To obtain these benefits, they discuss a novel configuration, proposed by Aplin, that has come to be known as a “split bar.” It consists of two large masses (two “ends” of a bar split in half) joined to each other by connection to either face of a layer of piezoelectric material. The benefit comes from the fact that, in this configuration, the piezo is actually functioning as the dominant spring in the system; by storing the bulk of the elastic energy, it is able to produce a larger amount of electrical energy. (In many ways this harks back to Weber’s original proposal to make the bar entirely of piezoelectric material.)

Gibbons and Hawking sketch the details of a detector of this sort, in which lead zirconate titanate (PZT) is substituted for crystalline quartz (used by Weber) because of its larger piezoelectric coupling constant. Then, in spite of the fact that the thermal noise power is increased because the piezo is a rather lossy spring, the sensitivity should be increased by more than a factor of ten in energy compared with Weber’s detector. At the same time, the optimum sampling time is shortened to 1 msec, so that more detailed information can be extracted from the signal.

In passing, Gibbons and Hawking also note that Weber uses a less than optimal way of searching for gravitational wave events. His definition of an event is a noticeable increase in the energy in the bar’s fundamental mode. But a gravitational wave impulse will only increase the energy if it arrives with a particular phase relationship to the bar’s previous excitation. If the wave arrives with a different phase, the bar’s energy may be decreased, or the effect may instead primarily change the phase of the bar’s vibration. They estimate that this means Weber saw only about l/4 of the events exciting a given bar. And, since the two bars being used for a coincidence have independent phases, only (l/4)’ = l/16 of the detectable coincidences would have been registered by Weber’s technique. This makes the question of the source of the gravitational luminosity that much more difficult to resolve. But it also means that, if Weber’s results were real, even more events should be detectable.

Whatever mysteries there may have been regarding Weber’s claims that he was detecting pulses of gravitational waves, if they were true they represented one of the most important astronomical and physical discoveries of the 20th century. So it is no surprise that a number of other workers decided to construct gravitational wave detectors. And quite naturally given Weber’s apparent success, most of these detectors were built quite

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Gaussian distribution of excitation expected on the basis of thermal and amplifier noise alone. The near-perfect Gaussian fit to their data then constitutes an apparent contra- diction of Weber’s results.

An important argument used to buttress the claim that the IBM detector was well understood was the application of electrostatic calibration forces to one end of the bar, and the successful detection of those events (within the statistical limits set by the signal- to-noise ratio) by the data processing system. Tyson’s 1972 presentation to the Texas meeting had previously emphasized the importance of this fundamental practice of experimental physics, as did the remarks of both Kafka and Drever. Levine and Garwin take Weber to task for having failed to use any calibration method, either as a check of his instruments’ front ends or of his data analysis procedure.

A more subtle implicit argument against Weber’s work is suggested to the reader of these papers by their admirable clarity, as contrasted with the rather Delphic pro- nouncements that fill Weber’s own contributions to Physical Review Letters. Levine and Garwin make this explicit at one point when they compare their results with their best guess at how Weber’s would be expressed in similar (sensible) units, complaining “We are thus forced to estimate these quantities, while noting that such information is easily obtained by the experimenter and is normally provided in the publication of a detection experiment.“46

Garwin led a crusade against Weber’s claims at the Fifth Cambridge Conference on Relativity (CCR-5), held at MIT on 10 June 1974.47 Among the topics discussed was 1) an error in the computer program used by Weber to identify coincidences, shown to generate nearly all of the coincidences in the one data tape shared by Weber with other researchers, and 2) the puzzling feature of Weber’s histogram of coincidences versus time delay showing a peak at zero delay in only the central 0.1 second wide bin, in spite of the fact that a 1.6 Hz wide bandpass filter was said to be part of the signal processing chain. But the most spectacular event of the discussion was what even those sympathetic to Garwin’s cause might have felt was a trick that bordered on unsports- manlike conduct. Weber had been given data from the detector of Douglass’s group at the University of Rochester, to search for excitations in coincidence with Weber’s own detectors; Weber reported at previous meetings that he had detected an excess of coincident events at a level of 2.6 standard deviations above the expected chance rate. According to Garwin’s account in a letter to the editor of Physics Today,47 “At CCR-5 Douglass revealed, and Weber agreed, that the Maryland Group had mistakenly assumed that the two antennas used the same time reference, whereas one was on Eastern

Daylight Time and the other on Greenwich Mean Time.” No stronger way can be imag- ined of impressing the community with the possibility that Weber was able, by some means, to find coincidences among any two data streams, whether the coincidences actually existed or not.

A panel discussion with almost precisely the same cast of characters as that of the 1972 Texas Symposium was staged at the 7th International Conference on General Rel- ativity and Gravitation in Tel Aviv, June 23-28, 1974.48 The plot, Weber’s lonely claims of detections contradicted by the null results of the others, was also unchanged-the only substantial difference is that Weber’s critics had had time to carry out more extensive searches and more careful data analysis. By this time, the Bell Labs group had carried out a coincidence run with an identical bar at the University of Rochester, operated by Douglass. The Munich group (which had by then incorporated the previously independent Frascati group) reported on the results of 150 days of coincident observations. Drever gave a report of a more extensive data run, seven months that had concluded in April 1973, yielding only one candidate coincidence; although this event could not be ruled out as a possible gravitational wave detection, neither could it be positively established as such in spite of the low probability that was estimated for it to have occurred by chance. (The detectors were only 50 m apart, and so may have both been driven by some other kind of influence.) In any event, Drever was able to show that the Glasgow experiment did not show the sort of event rate predicted by Weber’s experiment, except under rather implausible assumptions about the nature of the individual gravitational wave pulses. Tyson also briefly reported on the negative results from Garwin and from Braginsky.

How did the physics community deal with these contradictory results? This is an almost classic example of attempted replication of an important claim, but with both opposing camps standing firm in their beliefs that their own results were correct. Valu- able insight into the difficulties this situation posed to scientists can be found in the work of sociologist of science Harry Collins, who interviewed many of the principal actors during this period. His results are well worth consulting, even though the quotes from the interviews are reported without identifying the individual speakers.4g

The other key resource in the written record is the transcripts of the open discussions at the 1972 and 1974 panels. Both Kafka and Tyson point out strongly that Weber (usually) uses a far-from-optimal statistical method to look for signals. Tyson also comes close to accusing Weber of fraud; the method by which Weber has deluded himself and others is said to be continual “tuning” of the statistics used to search for

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“Although the non-existence [of Weber’s pulses] became obvious a long time ago, it still seems appropriate to publish our final negative result, because our experiment was as similar to Weber’s as possible, whereas all other coincidence experiments deviated in one way or the other.... Moreover, we think we have set the lowest upper limits obtained by Weber-type experiments over a reasonable long period of observation,” spanning 580 useful days of common observations of the two detectors. The main result of this paper is the null result that the statistics of the coincident excitation of the two detectors was just what would be expected from the laws of chance, given the noise levels in the detectors. Without mentioning Weber specifically, Kafka and Schnupp do remark that

Scanning our whole data, we could, of course, find periods of a few days, for which at some pair of thresholds the number of coincidences was up to more than 3 standard deviations higher than the average over the various time delays. However, the same was true for arbitrary delays, and zero delay did not seem to be distinguished in any obvious way. However, one should not forget: If one searches long enough in our finite sample of data, one must find some complicated property which distinguishes zero delay signficantly from the others. (Again this is true for an arbitrary delay, but with a different

propeW)

The paper goes on to pay special attention to two periods, totaling 67 days in length, when the operation of the Munich-Frascati experiment overlapped with times for which the Weber group claimed to have detected substantial rates of coincidences with its own detectors. The authors write: “These results do not give the slightest hint of a simultaneous influence on both detectors. If the significant observations reported by Weber’s group for these two periods had been due to gravitational radiation of any kind, they should have shown even more significantly in our experiment.” The mention of “any kind” of signals refers to the fact that the present authors used not only the vector- difference algorithm that is optimal for short pulses, but also used for these 67 days the algorithm preferred by Weber, which would be more sensitive for very long wave trains that gradually excited the antenna. Kalka and Schnupp conclude this section by remarking that “we do not have an explanation for Weber’s observations,” although they suggest the possibility that there might have been some undiagnosed electrical feedback from signals on the telephone line from Argonne into the Maryland bar itself.

The final section of the paper compares the likely strengths and rates of gravita-

tional wave signals from core collapse in supernovae with the then current and possible future sensitivities of gravitational wave detectors. In a dramatic figure, they superpose a model of the rate of supernovae at various distances from the Earth on the natural phase space for gravitational wave searches, event rate versus event strength. The authors point out that, even if one were able to improve the performance of gravitational wave detectors of the Weber type to the limit set by the Uncertainty Priniciple (by cooling, improving Q, or whatever other trick), one would still not have the sensitivity to detect events at the rate of several per year or greater. They conclude, “Because of the difficulties arising from this problem and because one would certainly like to measure more details than just the spectral density of pulses, the Munich group decided not to continue with (low temperature/high quality) Weber-type experiments, but rather with a Weiss-Forward type experiment, i.e., a laser-lighted Michelson interferometer.”

In spite of the considerations that moved the Munich group to abandon resonant- mass detectors, the groups that had decided in the early 1970s to build cryogenic versions of the Weber bar pushed ahead. A number of strong reasons can be given to justify this strategy, including the dubious value of relying (as the Munich group did) on signal-strength predictions which necessarily must be ignorant of truly novel astronomical phenomena, as well as the belief that evolutionary development is often a more rapid and reliable strategy for progress than a revolutionary approach. And, although progress was slower than the hopes expressed for it in Tyson’s 1974 remarks in Tel Aviv, this route did in fact lead to substantial increases in sensitivity well before the interferometric detectors began to catch up.

The first complete operating cryogenic resonant-mass detector was the one built at Stanford University by the group led by William Fairbank. In addition to the obvious reduction of thermal noise by cooling with liquid helium to a temperature of 4.3 K, and the use of the Josephson junction SQUID as a low noise preamp, there was another technical innovation that helped the Stanford bar reach a new level of sensitivity. This was the introduction by Paikz6 of a resonant transducer, tuned to the same frequency as the bar’s resonance, mounted on the end of the bar. Both Tyson and Garwin had used end-mounted transducers, but neither realized the advantages that would accrue to the tuned configuration-the ability to better “impedance-match” the mechanical excitation of the bar to the electrical system, thus increasing the coupling parameter /3. (See the discussion above.) The Paik transducer represented a new generation in another sense-it made no use of the piezoelectric effect, but instead used the motion of the resonant proof mass in the transducer to modulate the value of an inductance in

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whose detector wasn’t working as well as in 1981) or somewhat too early (from the point of view of LSU or Rome, who had not yet shaken all of the bugs out of their systems), there was another accidental side effect of its timing. That was the complete coordination of the time when all three systems went off the air to fix the problems that running together had made obvious. This is the reason that none of the state-of-the- art detectors was on the air on February 23, 1987, when Supernova 1987A appeared. The closest observed supernova in centuries was a chance no one would have chosen to miss, although in fairness at a distance of order 50 kpc it is unlikely, according to standard estimates of the gravitational luminosity, that it would have been seen.

There were, however, non-state-of-the-art gravitational wave detectors observing at the time. Weber has kept a room temperature bar in operation nearly continuously since the ’70s as has the Rome group. An unusual chain of reasoning was constructed, involving a suspect time for the supernova collapse, an unorthodox signature for the gravitational wave event, ad hoc assumptions about neutrino physics, and tremendous gravitational luminosity, but leading to a claim of significant detections of a large flux of gravitational wave pulses. 55 This claim has attracted much less attention than did the original claims of Weber in the early 1970s. A serious effort has been made to demonstrate that the statistical significance of the analysis has been overstated, due to construction of the signature to match the data stream.56

6 History of Interferometers

6.1 The Work of Gertsenshtein and Pustovoit

Almost as soon as Weber had begun work on the first gravitational wave detector of the resonant-mass style, the idea arose to use interferometry to sense the motions induced by a gravitational wave. Weber and a student, Robert Forward, considered the idea in 1964.” We will discuss below how Forward later went about implementing the idea. But the first discussion of the idea is actually due to two Soviet physicists, M. E. Gertsenshtein and V. I. Pustovoit. They wrote in 196257 a criticism of Weber’s 1960 Physical Review article, claiming (incorrectly) that resonant gravitational wave detectors would be very insensitive. Then, they make a remarkable statement justified only by intuition, that “Since the reception of gravitational waves is a relativistic effect, one should expect that the use of an ultrarelativistic body-light-can lead to a more effective indication of the field of the gravitational wave.”

Gertsenshtein and Pustovoit followed up this imaginative leap by noting that a Michelson interferometer has the appropriate symmetry to be sensitive to the strain pattern produced by gravitational waves. They give a simple and clear derivation of the arm length difference caused by a wave of amplitude h. Next, they note that L. L. Bernstein had with ordinary light measured a path length differences of 10-l’ cm in a 1 set integration time. The newly invented laser, they claim, would “make it possible to decrease this factor by at least three orders of magnitude.” (The concept of shot noise never appears explicitly here, so it is not clear what power levels are being anticipated.) They assume that one might make an interferometer with arm length of 10 m, thus leading to a sensitivity estimate of 10-14/& for “ordinary” light, or as good as 10-17/& for a laser-illuminated interferometer. This, Gertsenshtein and Pustovoit claim, is lo7 to 10” times better (it isn’t clear whether they mean in amplitude or in power) than what would be possible with a Weber-style detector. Putting aside their unjustified pessimism about resonant-mass detectors, their arguments about interferometric sensing are right on the mark, even conservative.

For improvements beyond the quoted level, they make suggestions that are somewhat misguided. They say that observation time could be lengthened beyond 1 set, which would be obvious for some sources (such as “monochromatic sinusoidal signals” or signals of long period) and hopeless for short bursts. Their other suggestion is to use “known methods for the separation of a weak signal from the noise background”; this suggestion is curious because known methods appear to be already built into their estimates that are referenced to a specific observing time. The other lack that is obvious in hindsight is any mention of mechanical noise sources. Still, the gist of the idea of interferometric detection of gravitational waves is clearly present, as is a demonstration that the idea can have interesting sensitivity.

6.2 The Origins of Today’s Interferometric Detectors

For a variety of reasons, not least of which must have been the fact that it was written too early (before Weber’s work had progressed beyond design studies), the proposal of Gertsenshtein and Pustovoit had little influence. The activity that began the by-now flourishing field of interferometric gravitational wave detection started independently in the West. In fact, it began semi-independently at several places in the United States at around the same time. The roots of this work can be seen in a pair of papers, written in 1971-2, by two teams linked in an unusual collaboration that is acknowledged in

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of the organization at MIT that administered the umbrella research grant supporting his work.” Weber’s claimed detection of gravitational waves was very much on Weiss’ mind in 1972, reported as possibly correct but with the recognition that the energy flux the waves appeared to carry would dominate the luminosity of the Galaxy. Weiss states that he had been inspired by a 1956 paper by F. A. E. Pirani (that discussed the identi- fication of measurable quantities in general relativity)g to consider the possibility that measurements of the light travel time between freely-falling test masses would make the best probes of spacetime structure. He further states that he had realized several years prior to writing (while teaching an undergraduate seminar) that the newly developed lasers could turn Pirani’s gedunken experiment into a practical measurement strategy. Weiss also notes that the idea “has been independently discovered by Dr. Philip Chapman of the National Aeronautics and Space Administration, Houston.”

Many of the ideas that appear in the breathless final paragraph of Moss et al. are elaborated at substantially greater length in Weiss’ report, which should be considered the first serious design study of the concept of interferometric gravitational wave detection. After the review of Weber’s claims, Weiss continues with a clear summary of the physical meaning of gravitational waves in general relativity, and an examination of the possible strength of gravitational waves from the then newly discovered pulsars. He then gives a summary of the key ideas of the proposed system:

l a Michelson interferometer used as a sensor of “differential strain induced in the ?lllllS,”

l operated “on a fixed fringe by a servo system” in a modulated system very much in the tradition of Dicke’s improved Eotvos experiment,58

l “mirrors and beam splitter mounted on horizontal seismometer suspensions” that “must have resonant frequencies far below the frequencies in the gravitational wave” and “a high Q,”

l arms that “can be made as large as is consistent with the condition that the travel time of light in the arm is less than one-half the period of the gravitational wave,” in part by being arranged as “optical delay 1ines”of the style described by Herriott.

Weiss is quite clear about the advantage that accrues from the last point. He says

This points out the principal feature of electromagnetically coupled antennas relative to acoustically coupled ones such as bars; that an electromagnetic antenna can be longer than its acoustic counterpart in the ratio of the speed

of light to the speed of sound in materials, a factor of 105. Since it is not the strain but rather the differential displacement that is measured in these gravitational antennas, the proposed antenna can offer a distinct advantage in sensitivity relative to detecting both broadband and single-frequency gravitational radiation. A significant improvement in thermal noise can also be realized.

This last sentence points out one of the key insights of this report, expanded upon at much greater length in the remainder of the text. As a sensitive mechanical measurement, the interferometric detection of gravitational waves is prey to a host of mechanical noise sources whose strengths need to be minimized if success is to be achieved. By far the largest section of the paper is devoted to estimates of the magnitudes of a long list of noise sources of various kinds. They include: amplitude noise in the laser (the only place where the work of the Hughes group is cited, as an example of a shot noise limited measurement), phase noise in the laser, mechanical thermal noise, radiation pressure noise, seismic noise, thermal gradient (“radiometer effect”) noise, cosmic ray impacts, “gravitational-gradient” noise, and fluctuating forces from electric and mag- netic fields. This looks almost (with a few omissions) like the list of noise sources that contemporary workers are grappling with as they strive to make the new kilometer scale interferometers work; by contrast, the other earlier treatments of the subject look myopic and unbalanced. And this insight is what led to the recognition that interferometers of the greatest practical length, with the resulting dilution of displacement noise terms as compared with a strain signal, would be the way to achieve the promise of good gravitational wave sensitivity, and would be worth the substantial investments needed to build them.

6.3 Interferometer as an Active Null Instrument

The agreement between Weiss and the Hughes group on the basic features of an interferometric detector must have something to do with the fact that they and Chapman were engaged in a remote three-way collaboration. But the fact that the key features of the design remain current to this day (with a few important additions) is evidence that they responded thoughtfully to an inherent logic of experimental design. Interfero- metric gravitational wave detection represents an extreme example of the application of design principles of wide validity in experimental physics. It is worthwhile to examine those principles here.

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to waste it dealing with noise outside the band in which one expects to find signals. Measures to deal with l/f noise often dominate the design of a high-sensitivity

experiment. No one explored these measures more systematically than did Robert H. Dicke, who was for a few years in the early 1960s Weiss’s mentor. Dicke was moved to think deeply about these problems while working on the development of microwave radar at the Radiation Laboratory at MIT during World War II. Devices called radiome- ters, receivers that measure the total power emitted by a broadband (often thermal) source of microwaves, would have had a variety of uses, if they hadn’t been rendered so insensitive by the large amount of l/f noise in the RF preamps. Dicke invented what came to be called the “Dicke radiometer” specifically to solve this problem.60 The heart of the scheme was a device to periodically (at 30 Hz in the original case) interrupt the flow of RF power from the antenna to the preamp, replacing it instead with a thermal source of radiation. At the “back end” of the instrument, the electronics were arranged to give a measure of the difference between the detected power from the antenna and from the reference load. The reason this defeats the l/f noise in the amplifier is that the comparison between measure and reference is made rapidly enough that the preamp’s output can’t wander much in the interval. Or, described in the frequency domain, the signal has been translated from DC up to a high enough frequency that the preamp’s noise is not dominated by excess noise with a l/f character.

Dicke’s invention had the immediate effect that microwave radiometry became practical even for sources with rather low antenna temperatures; this made a substantial contribution to the Rad Lab’s missio# as well as to the practice of radio astronomy.62 But the greatest impact came from Dicke’s realization that this modulation technique would have broad applicability, wherever l/f noise was a problem. This insight led him to invent the lock-in amplifier, a universal back end that can control the chopping of an experiment, calculate the difference in output between the “on” and “off’ states, and average the result to further reduce the noise. By now, “lock-in amplification” (also referred to as “phase sensitive detection”) has become a nearly universal practice in the fight against l/f noise.

A second classic measurement carried out by Dicke illustrates further insights in the battle against l/f noise. The test of the equivalence of inertial and gravitational mass carried out by Roll, Krotkov, and Dicke58 is considered one of the great examples of a null experiment. As championed by Dicke, 63 this term refers to a measurement where an answer of zero carries tremendous meaning. Precise equivalence of inertial and gravitational mass (or in other words a zero value for their difference) means that

gravity can be described by a metric theory. Null experiments play a special role for experimentalists as well as theorists, be-

cause an instrument that reads zero is immune to many of the sorts of problems that plague non-null measurements. Among these are calibration drifts and limited dynamic range of an instrument (whether from noise or from non-linear response.) Of course, turning a theoretical zero into an idea for an instrument that yields a zero output takes deep insight. One could argue that the torsion balance used in the improved Eiitvos experiment, whose motion would track the Sun’s if the aluminum and gold masses on opposite sides had differing ratios of inertial to gravitational mass, is among the most elegant instruments ever invented.

Maintaining the integrity of a null measurement takes insight that goes beyond the design of the front end of the experiment. For example, it would be a good idea for the null position of the test masses to be arranged to correspond to a null response from the sensor. Then, one can ignore (to first order) fluctuations in the drive level of the sensor (such as the light power in the optical lever), since zero is still zero even if it is multiplied by, say, 1 .Ol instead of 1 .OO. There are a variety of ways to create a null output from an optical lever at one particular operating point. One way would be to use a matched pair of photodetectors, placed so that the light beam falls equally on each detector when the balance is at the null position; as the beam moves to follow the balance’s motion, one photodetector receives more light while the other receives less, and a differential amplifier will reveal the motion. This method is essentially a DC technique.

Dicke’s team implemented a clever variation that let them make use of the advantages of a lock-in amplifier. A narrow light beam fell on a single photodetector, after passing by a wire of comparable width that cast a shadow on the photodetector. The wire was caused to vibrate from side to side by driving a current through it at the frequency of one of its “violin” resonances; as it did so, its shadow also moved from side to side across the light beam. If the beam were centered on the wire’s position, then the light received by the photodetector would increase equally due to the wire’s vibration to the left or the right. But if the beam were off center, then one direction of wire vibration lets more light pass than the other. So in the centered case the photocurrent varies only at twice the frequency of the wire’s vibration, while in the off-center case the current contains a component at the wire’s vibration frequency, whose amplitude and phase carries the information of the position of the light beam with respect to the wire. A lock-in amplifier converts the modulated signal into one at DC. The wire vibration can

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front-end amplifier, and a painstaking derivation of the shot noise. Mechanical aspects of the interferometer, which mainly determine performance at lower frequencies, get shorter shrift: test mass suspensions are described in a single paragraph, as neoprene and brass stacks of “the desired height” with “typical frequency of 10 Hz,” without any discussion of the mechanical transfer function or of the thermal noise of the suspension. The three paragraph section devoted to the “isolation system” gives information on both seismic isolation tables and the vacuum system enclosing the interferometer, and includes the following remark, almost in passing: “The vacuum system and isolation tables were designed so that after an initial checkout and operation with 2-m sections of aluminum irrigation pipe (8.5 m total interferometer pathlength), those sections could be replaced with longer sections (up to 1 km) with a substantial increase in interferometer-gravitational radiation-strain sensitivity for the same photon-noise- limited displacement sensitivity.”

A careful discussion of the calibration of the instrument and of its linearity is provided, although without including any but the most cursory details about the servo system (whose actuator was a PZT stack on which one of the mirrors was mounted) used to keep the interferometer in lock. One presumes that ignoring the behavior of the servo was justifiable on the assumption that its bandwidth was smaller than 1 kHz. Indeed, given the strategic decision only to consider the output of the interferometer at frequencies above 1 kHz, most of the omissions that strike a modem eye as surprising can be seen to make sense.

The discussion of the operation of the interferometer as a gravitational wave detector begins with a paragraph that will evoke much sympathy from present-day readers. It repeats the “ultimate plan” of operation at a remote site with long arms, but concludes with the remark, “The funding for this next move proved to be unavailable so we concluded the program by operating the system as it was, despite the high level of acoustic, electromagnetic, and vibrational noise from the other activities in the building.” Op- eration was evidently difficult, since as the Abstract notes, “The laser interferometer was operated as a detector for gravitational radiation for 150 h during the nights and weekends from the period 4 October through 3 December 1972,” a duty cycle of a bit over 10%. Environmental noise was a serious problem, and was taken seriously: a set of monitors of seismic, acoustic, optical, and electrical noises was installed, and a measure of their outputs was recorded along with the interferometer output.

Part of what made taking data so difficult was the decision to take advantage of the high bandwidth available; although the band below 1 kHz was abandoned as useless

because of high noise levels, the upper frequency cut-off was taken to be 20 kHz. This choice was never discussed in the paper+ne might have expected some sort of argument on astrophysical grounds that such high frequencies might contain signals, but it is just as likely that the cut-off was chosen to match the bandwidth of the “high quality stereo tape recorder” that was used as the primary data storage medium. Dealing with this much data was a tremendous burden, given the state of computer technology in the early ’70s. In fact, the data processing was performed almost entirely by listening to the audio tape--one section of the paper is called “Calibration of Ear.” (One 10 msec digitized chunk of data is shown in the paper, both in the time and the frequency domain.)

This method of data analysis was a clever solution to a vexing problem, and indeed continues to be a model for qualitative analysis and debugging of interferometers today. But it showed its weaknesses in what might otherwise have been the most interesting section of the paper, “Comparison of Data with Other Observers.” Here, Forward looks for coincidences between the unexplained events in his data set (not coincident with environmental signals in the monitor channel) and events in the resonant mass detectors that were in operation at the same time, at Frascati, Glasgow, and the Maryland group’s detectors at College Park and Argonne. In every case, Forward found no event in his detector at the time of a candidate event from another detector. He notes ruefully that “The one ‘distinctive signal’ reported by the Glasgow group occurred at 13 h 07 min 29 set GMT 5 September 1972, which was prior to the start of the Malibu data collection period.”

The data were of course most interesting for their comparison to the results reported by Forward’s former mentor Weber, since the latter was continuing to report coincident events between his various detectors. Forward notes that there were seven time blocks during which unexplained events in the interferometer occurred in close proxim- ity to Weber coincidences. However, he further states, “Both raw power and derivative power-squared digitized data plots digitized to O.l-set accuracy were obtained from the Maryland group and compared with the 0.2-set accuracy Malibu data. None of the audible Malibu signals fell within 0.6 set of a Maryland-Argonne coincidence.”

Fair enough, but what can be concluded from this lack of coincidences? Not much, according to Forward, since “It is difficult to compare the relative detection capabili- ties of the various antennas since their amplitude sensitivities, bandwidths, and signal processing techniques differ widely.” He goes on to state that “at the time one of the bar-antenna systems produced an event or coincidence corresponding to a gravitational-

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subtlety. Fabry-Perot cavities also appear to be less subject to excess noise from light scattered into unanticipated paths from mirror imperfections, a problem not suspected by either Weiss or Forward.

. Achieving the high sensitivities to which we now aspire requires vacuum of a quality much better than the Hughes interferometer. Pressures of lOme’ torr or better are required. The vacuum pipes are themselves much larger in diameter, due partly to the great care needed to keep scattered light effects at low levels. Scattered light also demands that baffles be properly placed in the interior of the pipes.

So it was probably going to take more than adding a few kilometers of irrigation pipe to the Hughes interferometer to detect gravitational waves with an interferometer. The realization that all of these features would be necessary was daunting, and caused the character of work on interferometers to change. Instead of quick demonstrations, it was considered necessary to try to engineer the variety of subsystems that high sensitivity would require. Instead of a device that Forward could honestly describe as a “gravitational-radiation experiment,” workers conceived of their apparatus as “prototype gravitational-wave detectors.”

6.5.2 Ron Drever’s Bag of ‘hicks

Ronald W. P. Drever was one of the leaders of the generation of experimenters who followed Weber, only to find no signals that matched his claims. Rather than build a faithful copy of Weber’s original bar, he chose to follow the path invented by Aplin (and publicized by Gibbons and Hawking) of the split bar, which maximized the bandwidth of the detector. When it became clear that much greater sensitivity would likely be required, he (like the German group) chose to switch to work on interferometers. His work in this period is again marked by an enthusiastic exploration of clever ideas. It is not marked, however, with many conventional papers in refereed journals. Instead, his most stimulating work is to be found in conference proceedings and lectures at physics summer schools.

Several of Drever’s most important contributions are described in the text of the lectures he gave at the NATO Advanced Study Institute on Gravitational Waves held at the Les Houches Center of Physics in 1982.65 Cast as an overview of the interferometric method of gravitational wave detection, it is dominated by an account of three crucial improvements on the basic scheme of Weiss, Forward et al. Each of these vari-

ations has come to play an important role in the design of the large detectors now under construction.

The Introduction gives an astoundingly brief account of the history of the field:

An obvious way one might consider detecting gravity waves is through the changes in separation of free test particles, and the idea of using optical interferometers for observing this has certainly occurred to many physicists: indeed one might wonder why so few searches for gravity waves have been made this way.

The work of the Hughes group is mentioned in passing as demonstrating that a simple gravitational wave interferometer could achieve the shot noise limit. Weiss’s work is referred to later as having contributed the idea of the delay line as an “important practical method for improving photon-noise limited sensitivity.”

Drever goes on to describe the optimization of the parameters of a delay line, from the point of view of shot noise reduction. He then remarks on a “practical difficulty” that “became apparent in early experiments at Munich and at Glasgow-the poten- tially serious effect of incoherent scattering of light at the multireflection mirrors or elsewhere in the system.” The interference between scattered light and light following the intended paths (which is non-stationary because the path followed by the scattered light can vary in length both on long and short time scales) proved to be a very trou- bling noise source in delay lines. The German group, Drever notes, proposed a way of modulating the laser light that would minimize the problem. Drever then suggests that “another approach would be to make the path traveled by scattered light equal to that of the main beam, and this may in fact be achieved if another type of optical system, a Fabry-Perot cavity, is used instead of a Michelson interferometer with many discrete reflections in each arm.”

The basic idea was that light traveling between parallel mirrors can be, in effect, trapped for many round trips, until it is either absorbed, scattered, or leaked out by transmission through one of the mirrors. Thus, such a cavity can play the same role as a delay line with its many spatially separate reflections. The classic Fabry-Perot cavity used flat mirrors, usually equivalent to one another, usually closely spaced compared with their diameters or with the diameter of the beam of light, and usually operated in transmission (that is, with the interesting light emerging from the mirror opposite to the one into which the light was injected). What Drever proposed was rather different: a pair of small mirrors (no larger than necessary to keep diffraction losses small), spaced

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from the bright port may be nearly as bright as the light entering the interferometer from the laser. (This insight represents a profoundly different “take” on the issue than can be found in Weiss’ work; he instead worried about optimizing the shot noise versus number of bounces on the assumption, good for short arms and poor mirrors, that substantial losses would eventually occur.) The light exiting the bright port is every bit as good as “fresh” light from the laser, so it seems a shame to waste it. Drever’s proposal is to arrange by an appropriate set of mirrors to redirect the used light into the interferometer, in coherent superposition with light arriving directly from the laser. (This has to be done using a beam splitter or other partially reflecting mirror of a carefully chosen reflectivity.) This arrangement has in effect made the whole interferometer into a single Fabry-Perot resonant cavity, whose back mirror is the Michelson interferometer, and whose input/output coupler is the partially-reflecting recycling mirror.

In principle, the advantages that could be achieved with this technique are quite large. Drever quotes rms shot noise in a search for 1 msec pulses of IO-“, far superior to what could be achieved without recycling. He also gives a diagram showing how the technique could be applied to an interferometer whose arms were made of Fabry-Perot cavities. The components necessary to sense and control the various internal degrees of freedom are drawn in with dashed lines, as an indication of the provisional nature of the design. In fact, more subtle schemes have had to be developed to implement such as system. But, given the quality of the mirrors available today (and the lack of commensurate progress in the power levels available from stabilized lasers), power recycling has been adopted as an essential feature of every large interferometer under construction today.

The other “possibility” described here is one “for enhancing sensitivity for periodic signals.” This one is again introduced in the pedagogically simpler delay line interferometer. And again, the aim is to find a way to make use of the fact that, with good mirrors, the light would not be significantly attenuated after it has spent one half of a gravitational wave period in an interferometer arm. A periodic gravitational wave persists (by definition) for much longer than one half period; why not find a way to accumulate a phase shift on the interferometer’s light for a much longer interval? The scheme proposed here does just that, by arranging for light that exits one arm after one half cycle of the gravitational wave to enter the other arm, where it stays for another half cycle. The light changes arms at the same time that the gravitational wave changes sign, at least for the signal frequency that matches the length of the interferometer. A partially reflecting mirror governs how long the light repeats this cycle before finally

exiting the interferometer. As with power recycling, Drever goes on to show how a similar effect can be achieved in an interferometer that uses Fabry-Perot cavities.

It has been shown recently that the scheme can actually be implemented in a much more elegant way, using a single partially reflecting mirror at the nominally dark port. In analogy with power recycling, this scheme (called signal recycling) can be thought of as forming a single large Fabry-Perot resonant cavity out of the interferometer, this one resonant at the frequency of the signal sidebands on the laser light that have been created by the action of the gravitational wave. 68 This version of the idea will almost certainly also find application in the next generation of large interferometers.

6.6 The Garching 30-Meter Prototype Gravitational-Wave Detec-

tor

The 1988 paper to which we referred above was the account by the group at the Max- Planck-Institut fur Quantenoptik, the successor to the bar group of the early ’70s. Shoe- maker et aZ.‘j’ provided a beautifully detailed account of the best-characterized interferometer prototype yet built. It can be thought of as the work that brought to fruition, on the meter scale at least, the ideas embodied in Weiss’s 1972 design study. Through the ’70s and ’80s a number of groups (including Weiss’s at MIT, Drever’s at Glas- gow and at Caltech, and Brillet’s at Orsay) worked in parallel with the MPQ group to develop prototypes of kilometer-scale working interferometric detectors.70 The MPQ paper makes a nice example, though, since it is an especially complete account of a well-functioning instrument. So for pedagogical purposes we let it here stand for the large body of work done worldwide through the 1980s.

The interferometer described here had test masses 30 meters from the beam splitter; light made 45 round trips, for a total light travel time in an arm of 9 ps. The folding of the optical path was achieved with a Herriott delay line. The interferometer was illuminated with an Argon-ion laser at X = 514.5 nm, capable of supplying up to 0.23 W to the photodetector (at a bright fringe) after all optical losses in the interferometer are included. The test masses consisted of simple glass mirrors with a radius of curvature 31.6 m; they were suspended from single-wire slings of free length 0.72 m, giving a resonant frequency of about 0.6 Hz. Each of these was in turn suspended from a metal plate hung from coil springs. This upper level of the suspension not only added isolation along the optic axis, but gave isolation in the other degrees of freedom that might cross-couple into the sensitive direction.

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tenna System,” submitted to the U.S. National Science Foundation in October 1983.75 (It has since its presentation been called the “Blue Book” because of the color of the cheap paper cover in which it was bound.) It was prepared primarily by Weiss and two colleagues at MIT (Paul S. Linsay and the present author), as the product of a plan- ning exercise funded by the NSF starting in 198 1. The report also contained a section by Stan Whitcomb of Caltech on Fabry-Perot systems (as a partial counter to Weiss’ emphasis on Herriott delay lines), as well as extensive sections written by industrial consultants from Stone & Webster Engineering Corporation and from Arthur D. Lit- tle, Inc. These latter contributors were essential, because this document contains, for the first time anywhere, an extensive discussion of the engineering details specific to the problems of the construction and siting of a large interferometer. The report was presented, by both Weiss’s MIT group and that of Drever at Caltech, at a meeting of the NSF’s Advisory Council for Physics late in 1983. While not a formal proposal, it served as a sort of “white paper,” suggesting the directions that subsequent proposals might (and in large measure did) take.

The first half of the report is devoted to the physics of gravitational wave interferometers. This section reads much like Weiss’ 1972 design study, except that many issues only touched on briefly in the first paper are here discussed at substantially greater length. In the 11 years that elapsed between the two documents, there had been real progress on several fronts. There are chapters on sources of gravitational waves, the basic physics of the response of a free-mass interferometer to a gravitational wave, a discussion of beam-folding schemes and a summary of the current prototype interferometers, and another extensive discussion of noise sources. The report is bracketed by an introductory section outlining a history of the field to 1983 and by a pair of appen- dices, one of which compares the quantum limits of bars and interferometers and the other showing why the interferometer beams must travel through an evacuated space instead of through optical fibers.

The main emphasis of the Blue Book was less a discussion of physics per se than it was a consideration of the practical aspects of the experiment as an engineering and construction project. The completely new material appears in the second half of the Blue Book, in the chapters summarizing the work of the industrial consultants. Weiss believed that the only significant impediment to achieving astrophysically interesting sensitivity was the expense of building an interferometer with long arms (the issue that had brought the Hughes group’s progress to a halt). The industrial study was undertaken with the aim of identifying what design trade-offs would allow for a large system to be

built at minimum cost, and to establish a rough estimate of that cost (along with cost scaling laws) so that the NSF could consider whether it might be feasible to proceed with a full-scale project.

Before such an engineering exercise could be meaningful, though, it was necessary to define what was meant by “full-scale.” The Blue Book approaches this question by first modeling the total noise budget as a function of frequency, then evaluating the model as a function of arm lengths ranging from 50 meters (not much longer than the Caltech prototype) to 50 km. The design space embodied in this model was then explored in a process guided by three principles:

l “The antenna should not be so small that the fundamental limits of performance cannot be attained with realistic estimates of technical capability.” This was taken to mean that the length ought to be long enough that one could achieve shot noise limited performance for laser power of 100 W, without being limited instead by displacement noise sources, over a band of interesting frequencies. The length resulting from this criterion strongly depended on whether one took that band to begin around 1 kHz (in which case L = 500 m was adequate), 100 Hz (where L = 5 km was only approaching the required length), or lower still (in which case even L = 50 km would not suffice). Evidently, this strictly physics-based criterion was too elastic to be definitive.

l “The scale of the system should be large enough so that further improvement of the performance by a significant factor requires cost increments by a substantial factor.” In other words, the system should be long enough so that the cost is not dominated by the length-independent costs of the remote installation.

. “Within reason no choice in external parameters of the present antenna design should preclude future internal design changes which, with advances in technology, will substantially improve performance.” This was a justification for invest- ing in a large-diameter beam tube, and for making sure that the vacuum system could achieve pressures as low as lo-’ ton:

In an iterative process, rough application of these principles was used to set the scope of options explored by the industrial consultants. Then at the end of the process, the principles were used again to select a preferred design. Arm lengths as long as 10 km were explored, and tube diameters as large as 48 inches. An extensive site survey was also carried out by the consultants. It was aimed at establishing that sites existed that were suitable for a trenched installation (which put stringent requirements

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quencies acoustic noise drove the interferometer via a variety of coupling paths through the injection optics as well as the test mass suspensions.

The proposal records substantial progress toward design of a full-scale interferometer. It states that the collaboration had adopted the Fabry-Perot beam-folding system. A preliminary design is presented in an appendix of the proposal. It envisioned use of 5 to 6 W of light at 514 nm from an Argon laser employed in a power recycled configuration. An elaborate schematic diagram gave a hint of the complexity of the servos necessary to control the large number degrees of freedom that need to be kept locked for such an instrument to function. These include lengths of the arm cavities, the separation of their input mirrors from the beam splitter, the location of the power recycling mirror, and the lengths of various “mode cleaning” resonant cavities used for spatial filtering of the laser beam. In addition to these lengths, control of a number of angular degrees of freedom also needs to be included. Four separate RF modulation frequencies are specified to drive these servos. Special features are designed into the main cavity locking servo so that the phase modulation can be injected with a small Pockels cell without the inevitable losses dominating the performance of the recycling system.

Another appendix describes an alternate optical configuration based on Herriott delay lines. It employed 86 cm diameter silicon test masses of 450 kg. The simplicity of the servos was listed as one of its major advantages. A “closed-path” variation of this design was also presented, in which light leaving one arm is injected into the other. This is like a single-interchange version of Drever’s system for improving sensitivity to periodic waves. Here it was employed mainly to relax the tolerances on matching the curvature of the large mirrors. (The virtues of this design have recently been explored again by the Stanford group.77)

On the engineering and site issues, there had also been some progress since the Blue Book study, mostly made by engineers at Caltech’s Jet Propulsion Laboratory, but this was not considered complete enough to highlight in the proposal. Instead, one of the first proposed tasks was to complete a preliminary engineering design. Nevertheless, a mature understanding had been achieved of what LIGO ought to be. This insight was expressed in a list of “Essential Features of the LIGO”:

1. “Two widely separated sites under common management.” Two sites had been a feature of Weiss’ earliest thinking, to allow coincidence observations to search for transient signals. The new feature was the commitment to truly have them man-

aged as a single entity, “to guarantee that two receivers of nearly equal sensitivity are on line simultaneously at two sites, with a high live time.”

2. “Arm lengths of order 4 kilometers at each site,” a slight scaling back of the 5 km considered previously, but still long enough to strongly dilute the effects of displacement noise.

3. “The ability to operate simultaneously several receiver systems at each site.” In a way, this was the most ambitious feature of the LIGO concept. In part it grew out of a kind of conservatism that was not clearly spelled out, but that was nevertheless real. The early LIGO interferometers, if they were not to be extremely risky extrapolations from known technology, were unlikely to have sufficient sensitivity to be assured of detecting astrophysical signals. Even if that weren’t so, the project would have had to wrestle with the competition between time devoted to observation and time devoted to improving the performance of the instrument. This competition had bedeviled workers on resonant-mass detectors. The key new idea for LIGO was that the precious commodity, an evacuated beam pipe, might be available with abundant cross-sectional area since the Fabry-Perot geometry had been adopted. All that was required was an arrangement of tanks at the ends of the pipe to install the test masses of various interferometers, both operational and experimental. This actually called for substantial cleverness in developing an airlock system, so that installation and operation could take place with “a minimum of mutual interference.”

4. “The capability for receivers of two different arm lengths.” Drever urged the adop- tion of this feature, to allow a clean test of the gravitational origin of candidate signals, which should show up as the tidal signature that a longer interferometer sees twice the signal.

5. “A vacuum tube diameter of order 48 inches.” This had the conservative justification that it would be necessary if one had to switch from Fabry-Perot cavities to delay lines, and the great benefit of allowing multiple FabryPerot interferometers, as mentioned above.

6. “The capability of a vacuum level of 10-a torr.” This would be needed, not for the first LIGO instrument, but to avoid having fluctuations in the index of refraction of the residual gas dominate shot noise in a more sensitive “advanced” receiver.

7. “A minimum lifetime of the facilities of 20 years.” This was to be not just a one-shot discovery experiment, but a laboratory to exploit the gravitational wave

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to guarantee detection of signals. This includes laser power of 60 W recycled by a factor of 100, a much more aggressive vibration isolation system, and final pendulum suspensions with a quality factor of 10’ carrying one-ton fused silica mirrors.

6.7.4 The Situation Today

Construction of LIGO was approved in 1991. By mid-1998 (the time of the writing of this review), construction of the two facilities in Hanford, WA and Livingston, LA was over three-quarters complete. The schedule calls for construction to be completed soon. Roughly speaking, 1999 is to be devoted to installation of the scientific equipment in the completed facilities, 2000 to shakedown of the interferometers, and 2001 to engineering activities to bring the performance up to the design specifications. Then, data will be collected during 2002-3. Beginning in 2004, upgrades to improve the performance will be carried out, interspersed with additional periods of observation.

The first instrument to be installed is expected to have a noise spectrum like that shown in Figure 4. The high frequency noise spectrum should be dominated by shot noise, as determined from an input power of 6 W, multiplied by a power recycling gain of 30. Thermal noise from the 1 Hz pendulum mode will dominate the intermediate frequency band; the oscillations of the 10 kg test masses should achieve a quality factor of 1.6 x 105. (Internal thermal noise will dominate the spectrum only in a narrow band, due to test mass modal quality factors of about 106.) Low frequency noise will be governed of course by seismic noise that passes through the multi-layer stack.

Performance of the VIRGO 3 km interferometer will be similar at medium and high frequencies. At low frequencies, seismic noise should be much lower in VIRGO than in LIGO, since a much more aggressive filter has been designed. This should allow the noise to be dominated by pendulum thermal noise down to 10 Hz or a bit lower.

The GE0 600 meter interferometer is not expected to reach quite such low levels, but it will be surprisingly close. To make up for the shorter length, advanced technolo- gies (including signal recycling) will be pursued aggressively from an early date. Thus this instrument will play a dual role as part of the global network of interferometers and as a prototype for features that will later be incorporated into other interferometers.

Frequency (Hz)

Fig. 4. An estimate of the noise spectrum of the LIGO I interferometers. The four most important noise sources are shown: seismic noise, pendulum mode thermal noise, thermal noise of internal vibrations of the mirror, and shot noise.

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