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General Relativity

The Michelson Laser Interferometer

A  gravitational wave passing through two objects at rest and floating free in space would cause those objects to move relative to each other. By measuring the changes in distance between these two objects, one measures the amplitude of the gravitational wave. The Michelson laser interferometer is a device that can measure this changing distance; but it suffers from the problem that we all had as children when we looked through a telescope for the first time: it keeps jiggling. The challenge is to design an interferometer that doesn't jiggle; gravitational wave have such a light touch that interferometer experiments must be able to measure fractional changes in length of 10-23. The frequency window for a ground-based interferometer is between 10 Hz and 1000 Hz. Neutron star binaries, black-hole binaries, spinning neutron stars, and supernovae are expected to be the sources of gravitational waves at these frequencies.

The basic design of a Michelson laser interferometer is to take a single beam of laser light, split the beam in two, sent the beams over paths that are at 90° to each other, reflect the beams back by mirrors at the end of each path, and combine the beams to produce an interference pattern.1 The original Michelson interferometer, which preceded the invention of the laser, was used to prove that electromagnetic radiation did not propagate through an ether, a result that later supported the theory of Special Relativity, the current theory of space and time. The current generation of Michelson interferometers is now testing the theory of General Relativity, the current theory of gravity (Much of what follows is based on the article by Sigg [1998];2 see also the article by Ju, Blair, and Zhau [2000]3).

The simplest design for the Michelson interferometer has a single suspended test mass at each light path, and light makes a single round-trip along each path. From the standpoint of ground-based instruments, this is intolerable, because an instrument sensitive enough to observe gravitational waves would have to be between 100 and 10,000 kilometers in length.

The solution to the size problem is to add secondary mirrors along each light path so that the light path is folded upon itself. The tradeoff with this design is that the size of the mirrors limits the number of reflections that are possible before the laser beam becomes broader than the mirrors, and the additional reflections degrade the sensitivity of the instrument.

A further improvement is introduced by adding a Fabry-Perot cavity onto each light path. In a Fabry-Perot cavity, one end of the light path is a fully-reflecting mirror, while the other end is a semi-transparent mirror. The distance between these two mirrors is actively held at an integer multiple of the light's wavelength. Laser light is introduced into the cavity through the semi-transparent mirror; the tuning of the cavity size creates a resonant wave pattern for the light. The practical effect of this complication to the instrument is that the effects of mirror size and multiple reflections on the instrument's sensitivity are greatly reduced.

One more complication is the addition of a mirror to reflect laser light that would exit the instrument back into the instrument.

The trick in making all of this work is the suppression of noise from a wide variety of sources. For Earth-based instruments, the primary sources of noise are earthquake, thermal noise, and shot noise. Earthquakes you know about; the instruments are isolated from the Earth by a suspension system, but the effect of earthquake cannot be eliminated, and they set the sensitivity of the instrument for gravity waves of frequency less than 100Hz. The thermal noise occurs from the thermal vibration of the atoms within the materials of the detector's suspension system. A longer path length for the light between the mirrors and a pendulum suspension system help eliminate this source of noise. The final source of noise, the shot noise, is a consequence of light being composed of individual photons. The distribution of photons are random, and so there are randomly-created bunches of photons in the detector, which adds an element of randomness into the measurement of the interference pattern produced when the light beams are recombined. For ground based gravitational wave detectors, these errors create a window between 10 to 1000 Hz over which waves can be detected, with the greatest sensitivity at 100 Hz. There are currently 8 ground based instruments either in operation or under construction. To date, they have not detected gravitational waves.

For a space based instrument, the design can be dramatically simplified, because distances of tens of thousands of kilometers are no longer an impediment if the reference masses of the interferometers are freely floating spacecraft. The earthquake noise is also absent, which allows a space-based interferometer to detect gravitational waves of much lower frequency. Going to space, however, introduces new sources of noises, such as the buffeting and electrical charging of the spacecraft by the solar wind, the pressure of solar radiation, and the gravitational pull on the spacecraft of Earth. The tradeoff gives an instrument that is sensitive to gravitational waves between 10-4Hz and 10-1Hz. Currently only one space-based interferometer is being funded, and this is the LISA experiment.

1 A laser interferometer measures small changes in length by creating interference patterns with laser light. The laser light beam is an electromagnetic wave that has a single wave pattern of a single frequency. If one were to split this light beam into two beams, let them traveling over different paths, and then recombine them, one would have an interference pattern that is dependent upon the difference in length of one light path relative to the other. If the difference in length is an integer multiple of the light's wavelength, then the two beams add constructively, and the result is equal to the original light beam. If the difference in length is an integer multiple plus one-half of the wavelength, then the beams add destructively, and the light intensity goes to zero. Any other length produces light somewhere in between in intensity. Observing the fluctuations of the intensity gives a measure of the changes in the relative path lengths of the two beams.

2 Sigg, Daniel. “Gravitational Waves.” in Neutrinos in Physics and Astrophysics: From 10-33 to 1028 (TASI 98), edited by Paul Lackgacker. Singapore, World Scientific, 2000.

3 Ju, L., Blair, D.G., and Zhau, C. “Detection of Gravitational Waves.” Reports on Progress in Physics 63 (2000): 1317–1427.

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