Theory - Interferometry

This week we will be learning about light, interference, and measurement. The full MOT experiment relies on a lot of engineering, including modules and devices that were studied and made in the last century! You will learn about interferometry, a measurement method that relies on the interference of waves of light.

Goals

By the end of this module, you should

• Know the importance of interferometry and its definition.

• Know what we can measure using intererometry.

• Know the constructions for the multiple interferometers.

• Know the differences between the Michelson and Mach-Zehnder interferometer.

• Recognize why we use interferometry for wave measurements versus direct measurements.

• Understand the Elitzur-Vaidman Bomb experiment and its connection to the power of superposition and quantum phenomena for measurements.

Deliverables

• Create a sketch of the many interferometers, annotated with the importance of the interferometer and the components of each of them.

• Draw a schematic for the Elitzur-Vaidman Bomb thought experiment, including calculations and/or intuitive explanations of all the outcomes.

Waves and Interference

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. At the most classical level, you’ve likely seen wave interference in bodies of water!

The 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

$$E_1 (t) = E_0 \cos(\omega t)$$

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

$$E_2(t) = E_0 \cos (\omega t + \Delta\phi)$$

The resulting amplitude measured is the sum of both waves

$$E(t) = E_1 (t) + E_2 (t) = E_0 (\cos(\omega t)+\cos(\omega t + \Delta \phi)$$

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

$$2d = m\lambda$$

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

$$2d = \left( m+\frac{1}{2}\right)\lambda$$

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.

The Mach-Zehnder Interferometer

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.

Figure 2 from Zetie et al. Figure for the Mach-Zehnder interferometer.

Concept Check: Using what you learned about the Michelson Interferometer, before continuing, try to derive the expression for the final combined electric field!

To start, let’s consider the interferometer without any sample present. Let’s look at the upper and lower path separately to see what phase shifts we pick up. Note that there are two detectors in our setup, so we need to calculate four phases, $\phi_{U, T}, \phi_{U, R}, \phi_{B,T}, \phi_{B,R}$, where $U$ and $B$ are the upper and bottom path, and $T$ and $R$ are the top and right detectors. Let’s call the lengths of the two paths $L_U$ and $L_B$. Let the thickness of the beam splitters be $t_{BS}$. It’s important to recognize that the orientation of the beam splitter matters! Some of the light will traverse through the substrate of the beamsplitter before hitting the mirror and getting reflected. The light will only pick up a phase $\pi$ from reflection if it traverses through the air and hits the mirror.

Let’s start with the upper path. We start by reflecting off the first beam splitter, picking up a phase $\pi$. We then reflect off the mirror to pick up another $\pi$. Up to the beam splitter, we also picked up a phase from traveling $L_U$ distance of $\frac{2\pi L_U}{\lambda}$. If you look carefully at the setup, notice that the beam travels through the substrate before reflecting. Thus, we pick up a phase $\frac{2\pi t_{BS}}{\lambda}$. If it goes to the top detector, after being reflected it will travel through the substrate again. Therefore, the phases for both detectors are

$$\begin{aligned} \phi_{U, T} &= 2\pi\biggr( 1 + \frac{L_U + 2t_{BS}}{\lambda}\biggr), \\ \phi_{U, R} &= 2\pi\biggr( 1 + \frac{L_U + t_{BS}}{\lambda}\biggr). \end{aligned}$$

Now let’s look at the bottom path. The light is first transmitted through the beam splitter, but it passes through the glass substrate to pick up a phase$\frac{2\pi t_{BS}}{\lambda}$. It is then reflected once to pick up another phase $\pi$. From traveling the full path, it picks up a phase $\frac{2\pi L_B}{\lambda}$. If it reaches the top detector, it travels through the beam splitter substrate again. If it reaches the right detector, it just picks up a phase $\pi$ from reflection. The total phases are therefore

$$\begin{aligned} \phi_{B, T} &= 2\pi\biggr( \frac{1}{2} + \frac{L_B + 2t_{BS}}{\lambda}\biggr), \\ \phi_{B, R} &= 2\pi\biggr( 1 + \frac{L_B + t_{BS}}{\lambda}\biggr). \end{aligned}$$

The phase difference in the detectors will be

$$\begin{aligned} \Delta_T &= \phi_{U,T} - \phi_{B, T} = 2\pi \frac{L_U - L_B}{\lambda} + \pi\\ \Delta_R &= \phi_{U,R} - \phi_{B, R} = 2\pi \frac{L_U - L_B}{\lambda} \end{aligned}$$

We can make our interferometer a bit more interesting by including a sample in the top path. We should note that this treatment of the interferometer was purely classical.

Concept Check: What is a Mach-Zehnder interferometer and how does it work? Draw a schematic for the interferometer and label all the paths and components.

*Concept Check: What exactly are we able to measure, both directly and indirectly, with the Mach-Zehnder interferometer? Mess around with the equations and the schematic above to see how knowing certain physical parameters would let you deduce other values. What if we placed a sample within one of the paths?

Concept Check: What are the main differences between the Michelson and Mach-Zehnder interferometer? What are the benefits and drawbacks of the two interferometers? Why would you pick one over the other?

Why are absolute wavelength measurements hard?

By now you might have noticed that the interferometry schemes can be used to measure the wavelengths of the used light. If we know the physical parameters of our setup, we can use the interference results to find the wavelength of our light source.

However, this is not a direct measurement. There really is no way to measure the wavelengths directly. This is due to the wavelength scale being extremely small and the quantum nature of light. The only ways we can measure wavelength are through interferometers, or using diffraction gratings

The Elitzur-Vaidman Bomb tester - a though experiment

Interferometers have been proven to be very useful in the present day. Aside from engineering feats, interferometers have also led to some interesting thought experiments. In 1993, Avshalom Elitzur and Lev Vaidman came up with the Bomb Tester thought experiment. It illuminated the power of the quantum properties of light, interference, and measurement. This thought experiment has actually been experimentally verified in the present day.

Here’s the experiment - Let’s say we once were responsible for making many, many bombs. We engineered these bombs to be detonated by light. However, years later, some of the bombs have become duds. Our job is to find the bombs that still work. The problem is, if we manually checked all the bombs one by one with a flashlight, not only would this be inefficient, but we would explode for every working bomb! Is there a way to determine which bombs are duds without blowing up our state-of-the-art lab?

The solution actually relies on the Mach-Zehnder interferometer, with a few twists! Up until now, all of our treatment has been fully classical. However, you might recall that light has wave-particle duality. The particle version of light is called a photon. Imagine this as a quantized packet of light with a fixed energy, wavelength, and frequency.

How does this change our Mach-Zehnder interferometer? With a proper treatment of quantum mechanics, you can redo the derivations using photons! Even without it, the intuition is the following - we can think about the outcomes probabilistically. A photon will hit the beam splitter and enter a superposition of traversing both the upper and lower path. When we look at the end detectors, we will only measure photon in the top or right detector, not both. That said, all of the interference principles will still apply!

Now, let’s set up the Bomb Tester experiment. We will set $L_U=L_B$. Note that in this case, we have,

$$\begin{aligned} \Delta_T &= \phi_{U,T} - \phi_{B, T} = \pi\\ \Delta_R &= \phi_{U,R} - \phi_{B, R} = 0 \end{aligned}$$

We will now place our bomb sample in the bottom path. Let’s consider all outcomes.

If the bomb is defective, the photon will bass through as if nothing happened. In this case, since $\Delta_T=\pi$, we have destructive interference. The probability of measuring a photon in the top detector is 0%. The right detector has $\Delta_R=0$, so we have constructive interference. The probability of measuring a photon in the right detector is 100%.

If the bomb is working, we have a few outcomes.

Outcome 1: No photon is detected. In one case, the photon can traverse the bottom path and it will be absorbed by the bomb. In this case, our detector will not measure any photons because it was absorbed en route. This also means our lab probably exploded.

Outcome 2: A photon was detected by the right detector. In this case, the photon took the top path and at the final beam splitter went right.

Outcome 3: A photon was measured in the top detector. In this case, the photon took the top path and at the final beam splitter was reflected up.

In all of our outcomes, we only measure a photon in the rightmost detector if the bomb works. Also in this case, we are able to measure a photon while still keeping the bomb intact. Thus, we have a way to check if our bomb is a dud or not! We simply set up the experiment as described and check if the rightmost detector click.

Now, there are clearly some issues with this. We would ideally like to keep our lab intact more than 50% of the time. That said, this experiment is a start! In fact, since the inception of this idea, the Elitzer-Vaidman bomb has been tested in a real experiment. Groups have also found ways to improve the 50% chance.

Concept Check: How does superposition and the quantum properties of light help us with measurement? Use the Elitzur-Vaidman Bomb as an example. Draw a schematic for the experiment and either with calculations or intuitive explanations, explain all the outcomes.

Concept Check: How might you improve the probability of your lab not exploding? Is there a way to do this?

Pre-lab questions and concept checks

• What is a Michelson interferometer and how does it work? Draw a schematic for the interferometer and label all the paths and components.

• What is a Mach-Zehnder interferometer and how does it work? Draw a schematic for the interferometer and label all the paths and components.

• What exactly are we able to measure, both directly and indirectly, with the Mach-Zehnder interferometer? Mess around with the equations and the schematic above to see how knowing certain physical parameters would let you deduce other values. What if we placed a sample within one of the paths?

• What are the main differences between the Michelson and Mach-Zehnder interferometer? What are the benefits and drawbacks of the two interferometers? Why would you pick one over the other?

• Why is a direct measurement of wavelength so hard?

• How does superposition and the quantum properties of light help us with measurement? Use the Elitzur-Vaidman Bomb as an example. Draw a schematic for the experiment and either with calculations or intuitive explanations, explain all the outcomes.

• How might you improve the probability of your lab not exploding? Is there a way to do this?

References

• Zetie et al., How does a Mach-Zender Interferometer work?

• MIT OCW 8.04 Lectures by Barton Zwiebac

  • Lecture on Interferometry

  • Lecture on the Mach-Zehnder Interferometer, Lecture on Mach-Zehnder and Superposition

  • Lecture on Elitzur-Vaidman bombs

• Demo with Interferometers

• Experimental demonstration of the Elitzer-Vaidman bomb: “Interaction-Free Measurement.” Paul Kwiat, Harald Weinfurter, Thomas Herzog, Anton Zeilinger, and Mark A. Kasevich. Phys. Rev. Lett. 74, 4763 – Published 12 June 1995

• MIT OCW 8.13-14: Michelson Interferometer Experiment

• PhysLibreTexts on Michelson Interferometers

• Lab Demo focused paper on Elitzur-Vaidman Bomb: Moraes, "The Elitzur-Vaidman bomb test"

• The original paper by Elitzur and Vaidman: Elitzur, A C., and Vaidman, L. "Quantum Mechanical Interaction-Free Measurements"