 LIGO's detection of gravitational waves required the construction of the most sensitive scientific instruments ever built. Let's examine in some detail how these instruments operate. LIGO stands for Laser Interferometer Gravitational Wave Observatory. The fundamental operating principle is as follows. A laser emits a beam of light with a wavelength of about one micrometer. The beam strikes a beam splitter. Which splits the beam into two perpendicular paths. At the end of each path is a test mass, which in principle is in freefall. The goal of the interferometer is to detect the motion of these test masses due to a gravitational wave. The test masses have mirrored surfaces that reflect the beams back to the beam splitter. This creates two optical paths with proper lengths sigma x and sigma y. If the test masses move, these lengths change. The beam splitter directs part of each reflected beam onto a photo detector, which generates an electrical signal proportional to the total light intensity. Variations in that intensity indicate changes in the optical path lengths. Here's a simple visualization of the interferometer in action. The beam splitter creates two copies of the sinusoidal optical beam, which reflect off the mirrors and are recombined into a single beam with two components. The relative positions of the peaks and valleys of the two components determine the optical intensity at the detector. Movements of the mirrors change the lengths of the optical paths. This changes the relative position of the peaks and valleys of the two beams, and this push leads to a change in the detected intensity. In this figure, the red and magenta curves represent one period of the two beams arriving at the detector. The blue curve is their sum. In normal operation, the beams are positioned to cancel each other out, producing a zero sum. When the mirrors move and the optical path lengths change, the sum fluctuates. This fluctuation is the signal that indicates the presence of a gravitational wave. Mathematically, the beams are sine waves with amplitudes A and minus A. They oscillate at optical frequency omega. The relative positions are determined by a phase phi, or phi, which is proportional to the change in path length. One beam is shifted by plus phi and the other by minus phi. The detected optical power is the time average of the square of the sum of the two beams. This works out to a constant times the sine squared of phi. Optical energy comes quantized as photons. The detector essentially counts photons over a short time window. The average number of photons per count, n, is proportional to the detected energy, and we write n equals a constant n sub m times the sine squared of phi. Photons arrive at random. So if, for example, n is 10, it is only on average that 10 photons will arrive in a single time window. For any particular window, the number of photons arriving will fluctuate about this average. These variations follow so-called Poisson statistics. So, over time, the photon count will vary about its average value. Since the fluctuations are due to the quantum nature of light, they're called quantum noise. Quantum noise fluctuations, which we denote by delta n sub q, vary as the square root of the average photon count. The signal we must detect in the presence of quantum noise is the small change in optical intensity due to small changes in the path lengths of the interferometer arms. These manifest as small changes in the phase phi. That signal, shown here denoted by delta n sub gw, is proportional to the phase change delta phi. Delta phi, in turn, is proportional to the change in proper length of the interferometer arms, delta sigma, divided by the optical wavelength, lambda. For a gravitational wave with amplitude 10 to the minus 21, the length change of the 4 kilometer arms is 4 times 10 to the minus 18 meters. That's less than one-hundredth the diameter of a proton. A change so small that it would seem hopeless to detect. And indeed, for decades, conventional wisdom was that it would be practically impossible to detect a gravitational wave. With an optical wavelength of about one micron, the resulting phase change is only about 5 times 10 to the minus 11. To have any hope of detecting a gravitational wave, our signal has to exceed the quantum noise level. Taking the square of the ratio of these, we find that 4 n sub m cosine squared of phi delta phi squared must be larger than one. Assuming cosine phi has its largest possible value of one, we end up with the requirement that n sub m must be greater than about 10 to the twentieth. N sub m is the average number of photons we would detect if the beams were oriented to produce the greatest optical intensity at the detector. The corresponding optical power in each interferometer arm is on the order of 10 kilowatts, a level that is impractical for a laser to operate at continuously. Key to LIGO's success are three technical enhancements illustrated in this figure. In addition to mirrored test masses at the end of each arm of the interferometer, partially mirrored test masses are placed near the beam splitter at the front of each arm. This creates so-called Fabry-Perot cavities, in which the beams bounce back and forth hundreds of times before leaving the cavity. On each round trip, another delta phi phase change is added. The result is that the effective length of each four kilometer arm is more than a thousand kilometers, greatly increasing the sensitivity of the instrument. Additionally, this causes the optical power to build up to a much higher level than what enters the cavity. Another enhancement is the addition of a system of power recycling mirrors. When aligned for greatest sensitivity, most of the light coming out of the interferometer arms combines at the beam splitter in a beam directed back toward the laser source. The power recycling mirrors reflect most of this back toward the beam splitter, from which the power is directed back into the interferometer arms. Power recycling, combined with the Fabry-Perot cavities, results in 750 kilowatts of power in the interferometer arms when the laser power entering the system is only 125 watts. A final enhancement is a system of signal recycling mirrors. This reflects the tiny interferometer signal back into the interferometer arms and can be adjusted to effectively tune the interferometer to enhance certain desired gravitational wave frequencies. There is one final practical issue to consider. An interferometer gravitational wave detector operates on the assumption that the test masses are in free fall. Of course, the LIGO test mass have to be suspended, while still effectively being in free fall in the direction of the interferometer arm. LIGO achieves this by suspending test masses at the end of a quadruple pendulum system. Let's look at the easier to analyze case of a single pendulum. A test mass, M, is attached to a pendulum arm of length L, which is suspended from some point on a fixed structure. Gravity pulls the mass downward with force MG, G being Earth's gravitational acceleration. Suppose the pendulum is displaced from the vertical by an angle theta, and assume theta is so small that the sine and cosine of theta are well approximated by theta and one, respectively. Then the linear displacement of the mass parallel to the ground is X equals L theta. The force of gravity now has a component parallel to the pendulum arm and a second perpendicular to it. The perpendicular force is MG theta, and this force acts to return the pendulum to the vertical. Newton's law of motion is mass times acceleration X double dot equals minus MG theta, which we can write as acceleration equals minus G over L times X. The constant G over L is equal to the square of 2 pi over the pendulum oscillation period, and the maximum value of X is the gravitational wave strength H0 times the interferometer length L. So the maximum acceleration of the mass due to pendulum displacement is H0L times 2 pi over the pendulum period squared. Previously, we saw that the maximum acceleration due to the gravitational wave is H0L times 2 pi over the gravitational wave period squared. The period of gravitational waves observed by LIGO is around one-hundredth of a second, while the period of its suspension system is roughly one second. So the acceleration due to a gravitational wave is much larger than the acceleration due to displacement of the suspension system. Therefore, although the pendulum-suspended test mass is not rigorously in free fall, it effectively is in the back and forth direction of the interferometer arm, since the acceleration due to the pendulum suspension is negligible compared to gravitational wave effects. To have test masses truly in free fall will require a space-based interferometer, and such systems are indeed planned for the future. These systems will be able to detect gravitational waves with much longer periods at much lower frequencies than LIGO. This will allow us to observe new classes of phenomena.