Optically Measure Nanometer Distances on the Dinner Table at Home

In this experiment, we make a Fabry Perot (FP) cavity, measure it using an educational spectrometer and model it using an online thin film calculator ( https://www.filmetrics.com/reflectance-calculator ).

The gap of the FP cavity is typically on the order of a few wavelengths, and it can in principal measure very small distances (The most sensitive interferometer can at its most sensitive state, will be able to detect a change in distance between its mirrors 1/10,000th the width of a proton! This is equivalent to measuring the distance to the nearest star (some 4.2 light years away) to an accuracy smaller than the width of a human hair. )

In optics, a Fabry–Pérot interferometer (FPI) or etalon is an optical cavity made from two parallel reflecting surfaces (i.e.: thin mirrors). Optical waves can pass through the optical cavity only when they are in resonance with it. It is named after Charles Fabry and Alfred Perot, who developed the instrument in 1899. Etalon is from the French étalon, meaning "measuring gauge" or "standard".

Etalons are widely used in telecommunications, lasers and spectroscopy to control and measure the wavelengths of light. Recent advances in fabrication technique allow the creation of very precise tunable Fabry–Pérot interferometers. The device is technically an interferometer when the distance between the two surfaces (and with it the resonance length) can be changed, and an etalon when the distance is fixed (however, the two terms are often used interchangeably).

The heart of the Fabry–Pérot interferometer is a pair of partially reflective glass optical flats spaced micrometers to centimeters apart, with the reflective surfaces facing each other. (Alternatively, a Fabry–Pérot etalon uses a single plate with two parallel reflecting surfaces.)

( see https://en.wikipedia.org/wiki/Fabry%E2%80%93P%C3%A9rot_interferometer )

When two partially transmissive mirrors are brought near contact, the interference fringes become visible under white light. This is because, the gap is now on the order of the 'coherence length' of the light.

The FP can be thought of as two mirrors bouncing light towards each other, and each time transmitting some portion of it with an accumulated phase. When all transmitted beams are in phase, it is constructive interference and a transmission resonance occurs. (see Figures).

The free spectral range (FSR) is defined as the wavelength separation between transmission resonances, and is a strong function of the gap distance. Also, depending on the reflectivity of the half-silvered mirror, the resonance will have a finite quality factor, which is typically quantified by the Finesse of the cavity.

Here, we make a FP cavity using easy to find cheap parts, demonstrate the basic features of a FP cavity and measure its transmission using a low-cost spectrometer. By modeling it through an online reflectance calculator, we can fit the data and tell what the gap between the mirrors is.

Supplies

Helper Hands

Half Silvered Acrylic mirror( e.g. 12 x 12 x 0.04 Inch Acrylic See-Through Mirror, Scratch Resistant, 40% Transparent)

White light source (e.g. Maglite Xenon bulb)

Arduino UNO as voltage supply (3.3V output)

Jumper Cables

A spectrometer (e.g. Spectryx Blue)

Binder clips

Scissors or hobby knife

Step 1: Make the Fabry-Perot Interferometer

The half-silvered mirror has a blue coating sheet that can be carefully removed to expose the silvered surface(see figure).

Cut the mirror into 3 cm x 1 cm rectangles, remove the protective blue layer. Use binder clips to clamp the silvered surfaces. Make sure the silvered surfaces are facing each other.

If it is done right, the colored fringes show up between the ends of the binder clips. These fringes will be irregularly shaped because of the distortion of the acrylic mirrors, and they will be quite sensitive to bending or touching. You can observe this sensitivity by holding it against a light source and slightly applying pressure to the clips.

Step 2: Make a Transmission Setup

Use the helper hands as shown in figure to hold the white light bulb.

Connect the bulb to the Arduino 3.3V output and GND.

Hold the fiber from the spectrometer with the other crocodile of the helping hands.

Adjust the fiber to bulb distance to about 3 cm.

Start the GUI of the spectrometer and record a reference spectrum. Then switch to transmission mode.

(see for example https://www.instructables.com/AcidBase-Test-Using-Red-Cabbage-and-a-Spectrometer/ on realizing the transmission setup)

Step 3: Measure the Transmission of the Fabry-Perot Cavity and Model Response

Position the sample in between the light source and the fiber. Make sure the fiber almost touches the sample, this will give you sharper fringes.

Measure the transmission at different locations along the fringes. In the first figure, we see from top to bottom, three different locations starting from close to center towards outer parts. The sample is clamped and touches at the center, and because of a tiny curvature of the mirrors, the gap increases as we move further away.

We can use the online reflectance calculator to calculate the transmission. Through trial and error, by changing the air gap in the calculator, we can observe the fringes move (make sure the calculator is in transmission mode),

For the three positions, we predict gaps of 900 nm, 2264 nm and 2800 nm.

Finally, the data is downloaded from the calculator web page and plotted on top pf measurement results.

The agreement is not perfect, but acceptable. We don't actually know the silver thickness, or if the acrylic mirror is perfectly transmissive in the UV part of the spectrum. Unknowns such as these make the observations and calculations deviate from each other, especially in the UV part of the spectrum.

We have measured nanometer sized air gaps on the kitchen table, using easy to find tools!