# Theory - Interferometry (Michelson & Mach-Zehnder)

### Interference of light

All the phenomena studied in this lesson results from a physical phenomena called wave interference. When two coherent waves overlap one another, they either "interfere" destructively (subtractive), or constructively (additive). In this lab, we will force light of a single wavelength (coherent) to interfere, and observe/study this interference.

### Michelson Interferometer

The Michelson interferometer is a precision instrument that involves the setup shown above. Light coming from a laser/light source is split by passing it through a half-silvered mirror, transmitting half the intensity, and reflecting half the intensity. The transmitted light hits a movable fully reflective mirror, and the reflected light hits a fixed fully reflective mirror. Both beams are then reflected, overlapped and hit a detector.

As described, whether the beams interfere **constructively** or **destructively** depends on the relative phase of each of the combining beams at the detector. This is determined by the phase difference of the light. For path 1, we can describe the wave hitting the detector using

where $E_0$ is the wave amplitude, $t$ is time, and $\omega$ is the angular frequency of the light. For path 2, the wave will have a relative phase $\Delta\phi$ to the first wave which we call phase difference

The resulting amplitude measured is the sum of both waves

With **constructive interference **$(\Delta\phi=0)$, the wave amplitudes adds in such a wave to produce a maximum intensity beam striking the screen. The condition for maximum constructive interference is

where $m$ is an integer, and $\lambda$ is the wavelength of the incoming light. When the path length difference is an integer multiple of the wavelength, the recombining beams will be *in phase*, and the resulting amplitude of the combined beams is then the sum of the amplitudes of each beam.

With **destructive inteference **$(\Delta\phi = \pi)$, the phases of the light beams are such that the recombining beams cancel each other out. The condition for maximum destructive interference is

When the path length is an odd half integer multiple of the wavelength, the recombining light beams will be exactly out of phase. The resulting difference of the combining beams will be the difference of the amplitudes of each beam. If the amplitudes of the split beams are equal, the combined light beam will have zero amplitude.

#### LIGO fun-fact

The Michelson interferometer underpins the basic technology used at LIGO (Laser Interferometer Gravitational-Wave Observatory) operated by Caltech and MIT, where each arm of the interferometry spans more than 4km long! They have made significant changes to their design, including using (1)what we call Fabry-Perot optical cavities to multiply the path length experienced by the laser beams by 300X; (2) higher powered lasers; (3) exceptional upgrades to maintain the stability of the system, which enables them to achieve phenomenal sensitivity to the path difference, such that they are able to detect and measure* gravitational waves, *which changes the relative distance between the two interferometer arms by less than the width of a proton.

### Mach-Zehnder inteferometer

The Mach Zehnder interferometer is an adapted version of the Michelson interferometer for studying the change in the wavefront when the light wave passes an object of interest. An illustrative diagram is shown below

The light from a laser soruce is divided by a beam splitter into two beams of equal intensity. After reflecting from two mirrors, the two beams are recombined, and the inteference pattern can be observed on a viewing screen in either path.

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