Physics · Optics · Year 1

Interferometers and LIGO

How can light detect a shift hundreds of times smaller than the diameter of a proton?

The situation

About 1.3 billion years ago, in a distant galaxy, two black holes circled each other and merged. On 14 September 2015 the gravitational wave from that event passed through the Earth and stretched and squeezed the 4 km arms of the LIGO detector in the USA by a distance hundreds of times smaller than the diameter of a proton. The instrument that sensed it is a giant descendant of the tabletop interferometer that Michelson built in the 19th century. How can light waves become such a delicate “ruler”?

Topic

Interferometers. Michelson interferometer and LIGO: detecting a gravitational wave

Key ideas

Coherence
Interference fringes
Gravitational wave

Formulas

I = \frac{I_{0}}{2} (1 + V \cos \frac{2 π Δ}{λ}), Δ = 2 d \cos θ

Two-beam interference: $Δ$ — path difference, $d$ — difference in the mirror distances, $V$ — visibility. A bright ring appears at $Δ = m λ$ .

N = \frac{2 Δ d}{λ}, N = \frac{(n - 1) l}{λ}

Counting fringes: when the mirror moves by $Δ d$ (Michelson), or gas is pumped out of a cell of length $l$ (Jamin), $N$ fringes pass through the centre.

l_{k} = \frac{λ^{2}}{Δ λ}

Coherence length: if the path difference exceeds it, the fringes disappear. A He–Ne laser ≈ 20 cm, an LED ≈ 20 µm, white light ≈ 1 µm.

T = \frac{1}{1 + F \sin^{2} (δ / 2)}, F = \frac{4 R}{{(1 - R)}^{2}}, ℱ = \frac{π \sqrt{R}}{1 - R}

Fabry–Pérot: $δ = 4 π d \cos θ / λ$ , $ℱ$ — finesse. When $R$ is large, the rings are narrow.

h = \frac{Δ L}{L}, Δ φ = \frac{4 π N_{r} h L}{λ}

Gravitational wave: $h$ — relative strain. In LIGO $L = 4$ km, $λ = 1064$ nm, and in the resonators the light is reflected $N_{r} \approx 280$ times.

Theory

The Michelson interferometer

A beam splitter divides the laser beam in two: one part is reflected from mirror $M_{1}$ , the other from mirror $M_{2}$ , and they recombine on the screen. If the path difference $Δ = 2 d$ equals a whole number of wavelengths, the waves reinforce each other; if it equals a half-integer number, they cancel. Because of the diverging lens the rays leave at different angles: fringes of equal inclination (circular rings) appear on the screen. When the mirror moves by $λ / 2$ , one ring passes through the centre, so a displacement can be measured by counting fringes: $N = 2 Δ d / λ$ . If the mirror is tilted, straight fringes of equal thickness appear instead of rings. In 1892–1893 Michelson measured the standard metre in wavelengths of the red line of cadmium.

Coherence

Fringes are seen only when coherent waves — waves with a constant phase difference — are superposed. A real source has a spectrum of width $Δ λ$ : as $Δ$ increases, the fringes of different wavelengths shift apart and finally blur together. So fringes are visible only for $Δ ≲ l_{k} = λ^{2} / Δ λ$ : with white light just a few coloured fringes, with an LED tens of micrometres, with a laser tens of centimetres. The two close lines of a sodium lamp (589.0 and 589.6 nm) give “beats”: the fringes blur every time the mirror moves by 0.29 mm. The same two-beam interference works in thin films and Newton's rings (in detail — in the “Interference and diffraction” experiment).

The Michelson–Morley experiment

In 1887 Michelson and Morley tested whether, if the Earth moves through the supposed “aether” at 30 km/s, light would travel along one arm more slowly than along the other. The apparatus stood on a stone slab floating on mercury, and mirrors made the light path 11 m long. Aether theory predicted a fringe shift of $Δ N = 2 L v^{2} / (λ c^{2}) \approx 0.4$ when the slab was turned through 90°, but the observed shift was at least 20 times smaller. The speed of light does not depend on the direction of motion — this is one of the foundations of the special theory of relativity.

Fabry–Pérot and Jamin

In the Fabry–Pérot interferometer, light is reflected many times between two parallel half-silvered mirrors. The superposition of many beams makes the rings very narrow: at $R = 0.9$ the finesse is $ℱ \approx 30$ — the ring width is 30 times smaller than the spacing between rings. So it can resolve close spectral lines ( $δ λ = λ^{2} / (2 d ℱ)$ ) and serves as the resonator in lasers. In the Jamin interferometer a thick glass plate splits the beam into two parallel beams; they pass through two cells and recombine at a second plate. If gas is pumped out of one cell, the optical path decreases by $(n - 1) l$ and the fringes shift — this is how the refractive index of a gas is measured. Model: absorption in the mirrors and phase shifts on reflection are not taken into account.

LIGO and gravitational waves

When massive bodies accelerate, spacetime “ripples” — a gravitational wave travels at the speed of light. As it passes from above, one direction is stretched and the perpendicular direction is squeezed: a ring of test bodies becomes an ellipse. LIGO is a Michelson interferometer with 4 km arms; a Fabry–Pérot resonator in each arm reflects the light ≈ 280 times and lengthens the path to ≈ 1 120 km. The detector works at the dark fringe: if the arms are equal, almost no light reaches the output, while the smallest difference gives a signal. At $h = 10^{- 21}$ , $Δ L = h L = 4 \cdot 10^{- 18}$ m — about 400 times smaller than the diameter of a proton. If two detectors about 3 000 km apart (Hanford and Livingston) see the signal almost simultaneously, it is not local noise. Model: the wave arrives from directly above, with the most favourable polarisation; the signal shape is approximate and noise is not taken into account.

Sources and other telescopes

Sources of gravitational waves are pairs of merging black holes and neutron stars; spinning asymmetric neutron stars (pulsars) are expected to emit continuous waves, and pairs of white dwarfs in the Galaxy to emit low-frequency waves — the space detector LISA is planned to look for them. On 17 August 2017, almost at the same time as the GW170817 wave from a neutron-star merger, gamma-ray telescopes also recorded a flash. Interferometry works in radio astronomy too: telescopes a distance $B$ apart resolve an angle $θ \approx λ / B$ — the Earth-sized “Event Horizon Telescope” imaged the shadow of the black hole in galaxy M87 in 2019. X-rays and gamma rays are absorbed by the atmosphere and are observed by orbiting telescopes; neutrino and cosmic-ray detectors are covered in other experiments.

Did you know?

In the 1887 Michelson–Morley experiment, aether theory expected a fringe shift of 0.4, but what was observed was less than 1/20 of that. In 1907 Michelson became the first American to receive the Nobel Prize in Physics.
On 14 September 2015 LIGO recorded the first gravitational wave: black holes of about 36 and 29 solar masses merged, and energy equivalent to ≈ 3 solar masses was radiated as waves. For this discovery R. Weiss, B. Barish and K. Thorne received the Nobel Prize in 2017.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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Luminous flux and illuminance — Grade 11
Infrared, UV and X-rays — Grade 11
Scattering of light — Year 1
Crystal optics — Year 1
Resolving power — Year 1
Nonlinear optics: the green laser — Year 2