Fabry-Perot Spectrometer
Developing an inexpensive, high-precision tool to measure plasmas.
The light emitted from a plasma contains information about its temperature, density and bulk velocity. For example, the precise color of a specific emission can tell you how fast a plasma is flowing. If the plasma is flowing away from the where the light is collected, the color will be slightly redder than it would be if it wasn’t flowing (or the light will be bluer if it is flowing toward the view). This is called the Doppler effect; this is the same phenomena experienced when a siren changes pitch as a fire truck speeds by. In a similar way, the temperature of the plasma smears out a particular color because temperature is random velocity of individual particles.
In relatively low-temperature plasmas, such as the ones in PCX (the experiment I worked on at the University of Wisconsin-Madison), ion temperature can be difficult to measure. Ion temperature is a very important parameter of the plasma for analysis (plasma's viscosity has a very strong dependence on the ion temperature, for example). Usually, the Doppler broadening of these plasmas (ion temperature is <1eV ~ 11,000°C) is so small that it is difficult to observe with a standard spectrometer. Confronted with this problem, our lab needed to develop an inexpensive, high-precision spectrometer.
A standard spectrometer consists of a grating, which acts like a reflecting prism. When light is shined on the grating, it is reflected with the constitute wavelengths (colors) spread out in space. Unlike a prism, a grating achieves this wavelength separation by interference, which drastically increases the contrast of the resulting spectrum. If you know the precise details of the grating, you can put a detector in a certain location to measure the amount of light hitting that location, which you know is a certain wavelength. Grating spectrometers are great for general use, but have a glaring problem that in order to increase their resolving power (distance in space between different wavelengths), a larger grating is needed which is expensive and difficult to align. Instead, a Fabry-Perot spectrometer consists of two parallel, highly reflective mirrors, called an etalon, that light is shined into. The light then bounces back and forth between the mirrors many many times and exits with a ring-like pattern. Each ring represents a different wavelength, such that a smaller ring is slightly redder than a larger ring. By capturing an image of the Fabry-Perot rings, the entire spectrum of incoming light can be analyzed.
I worked with other graduate students and a French collaborator to design, build and implement a Fabry-Perot spectrometer for use on PCX. The design was constrained by the etalon that we managed to get from an emeritus professor and the need for large throughput of light. Initially we used a Nikon DSLR camera to image the rings, but ultimately the consumer camera sensor wasn’t sensitive enough for our needs. It was a fun learning experience to figure out how to control the Nikon via a python script communicating over a USB connection. Additionally the optical train went through several iterations: from a large optical breadboard taking up a 6x4’ table to a portable suitcase-sized version that traveled to Lyon, France. In the end the setup takes advantage of a large fiber optic bundle to avoid needing beam steering mirrors and is only about 1 meter long.
Fabry-Perot spectrometers aren’t widely used like gratings because, while the setup and alignment is fairly straightforward, the analysis is very challenging. The ring pattern output shows many (in our case 3-5) interference orders. An order is simply a repeat of the spectrum at a slightly different location. Grating spectrometers have orders as well, but it is easy to focus on the first or second one. For the Fabry-Perot, the innermost ring is actually the highest order number and the exact number is difficult to extract. The orders are determined by integer number of wavelengths of light that can fit into the gap between mirrors.
To determine which integer order you are observing, one needs to know how many times the light has bounced back and forth between the mirrors. Within a given order, the radius of a ring informs the wavelength of light, but this relationship changes slightly depending on the exact order number. Determining this order number is essentially a difficult minimization problem where there are many local minima, but a slight envelope determines which one is the global solution. In order to solve this problem consistently, we developed a Bayesian approach using multi-modal sampling that can determine the uncertainty in the order number (i.e. calibrate) using a spectrum taken from a thorium lamp. Once calibrated, the Fabry-Perot can be aimed at a plasma and the ion temperature and velocity can be determined with a given probability.
This project was a continual learning process. Throughout it I learned about advanced optics, Bayesian statistics, camera firmware control and a little bit of French. The result of all this work was published in Review of Scientific Instruments; the paper was awarded an editors’ pick for its issue.
If you are interested in more details, the paper is available here.