### The Fabry-Perot Interferometer

### Design of the Fabry-Perot Interferometer

### Special Characteristics of the Fabry-Perot Interferometer

### The Fabry-Perot Interferometer in Action

The Fabry-Perot interferometer, designed in 1899 by C. Fabry and A. Perot, represents a significant improvement over the Michelson interferometer. The difference between the two lies in the fact that the Fabry-Perot design contains plane surfaces that are all partially reflecting so that multiple rays of light are responsible for creation of the observed interference patterns. The general theory behind interferometry still applies to the Fabry-Perot model, however, these multiple reflections reinforce the areas where constructive and destructive effects occur making the resulting fringes much more clearly defined. This, as will be discussed later, allows for much more precise measurements of wavelength, and free spectral range.

The basic setup of the Fabry-Perot interferometer maintains the same general principle of the Michelson interferometer. It consists of two flat, planar, surfaces that are almost, if not exactly, parallel (remember that circular fringes only occur when these plates are parallel!). The distance between these partially reflecting surface may be changed as well, just as with the Michelson interferometer. This characteristic actually represents the reason for deeming it an interferometer, for if the plates were immovable, it would be referred to as an Etalon. These differences can be seen diagramed below:

Note that for the interferometer, the source is a broad one, while for the etalon it is a point source. In the interferometer picture (bottom), one of the planes will move.

If the plates of the Fabry-Perot device are moveable (i.e. it is an interferometer), then the fringes will move across the surface due to the change in constructive and destructive interference lengths. A photodiode, or photoelectric device, may be placed at the center of the interference fringes which will count the fringes as they pass by. If the distance the plate has moved by is precisely known, then it is possible to calculate the unknown wavelength of some source by order of interference relationship . The variable n is the order of interference, or the number of fringes that pass by in our case. L represents the distance traveled by the mirror.

Another unique characteristic of Fabry-Perot Interferometer is the Free Spectral Range. This also has to do with fringe frequency, and is defined as the separation between adjacent orders of interference according to Fowles. This definition refers to how far the moving mirror must move to arrive at the next order of constructive interference which occur every . This relationship says that orders of constructive interference occur every half-multiple of the wavelength of the studied light.

Finally, the resolving power of the Fabry-Perot device represents another innovation over the standard Michelson interferometer. The resolving power may be defined as the ability to differentiate between adjacent fringe systems. Imagine for an instant, that a photodetector device is placed so that it is detecting the fringes that go by while one of the reflective, planar surfaces is being moved. This change in fringe maxima can be modeled mathematically by what is known as an Airy function in the form . The delta term represents the length for a change in order of constructive interference and occurs in integer multiples of pi. The reason why half of this distance is chosen is because that satisfies the Taylor criterion for the definition of resolution. According to these standards, two lines are considered to be resolved if the individual curves of the Airy function cross at the half intensity point (occurs at half the length of order of constructive interference). Thus, if attempting to resolve two sets of fringes, the total intensity observed will be represented as the sum of two Airy Functions as follows:

Here, F is termed the coefficient of finesse and is equal to . The R term represents the reflectance of the planar surfaces in the Fabry-Perot device. When F is small, the fringes will appear smeared and indistinguishable. We see these effects using Mathematica below:

The above represents the general behavior of the Airy function as R is increased or decreased (sample for HeNe laser wavelength). Clearly, the fringes are thicker and less defined as R decreases (Picture on right). Since the clarity of the fringes depends on the reflectance of the surfaces involved, I will go ahead and handwave some nasty math and have you trust that the resolving power will also depend upon the reflectance coefficient R. Sure enough, the resolving power of the device does indeed depend on the reflectance coefficient in the manner . The resolving power basically says that for a given value of mirror separation (remember n is the order of interference and depends upon separation length of the mirrors!) the resolving power can be increased infinitely as the reflectance goes to unity. Remark, however, that nature doesn't like infinity so perfectly precise resolution will never be achieved. The reflectance term ends up limiting the resolving power of the device because mirrors that use silver or aluminum coatings on the planar surfaces are actually only about 80-90 percent reflective. Thus, absolute resolution will never be reached due to physical constraints.

The Fabry-Perot interferometer also represents a tool essential to studying lasers. Laser output isn’t truly monochromatic due to the phenomenon of Doppler spreading. Consider a source and observer of light moving relative to each other . If the observer moves away from the light source at a speed much less than c, the frequency (f'') of the approaching light appears to be shifted from the expected frequency (f) according to the relation f'' = f (1-v/c). As a result, atoms of different velocities will absorb and emit photons of slightly different frequencies from the same energy line. In a gas, atoms have a broad range of velocities, so the output peaks of a gas laser will be broadened. Ideally, the emission profile should be a sharp peak, but due to Doppler broadening it appears Gaussian. Although the gas emits radiation that has a Gaussian lineshape, the laser cavity does not permit a continuous spread of frequencies to resonate. Only those frequencies that achieve constructive interference in the cavity will reach the threshold gain level and lase. When this constructive interference occurs, the length of the laser cavity will be an integer number of half-wavelengths of the resonating frequency where n represents an integer greater than zero. This expression can be solved for the permitted frequencies . Thus, the difference between adjacent permitted frequencies is .

In the experiment performed by Dr. Von Seth Carpenter and Jimmy Jim Nolen, they were able to measure the frequency separation between the longitudinal modes of a HeNe laser. They noted that the Free Spectral Range was 2 GHz, according to the manual accompanying the particular Fabry-Perot Interferometer that they used.

For a cavity length of 46.9 cm the frequency separation between modes was measured to be325.5 MHz. Theoretically, the separation should be . The Doppler width of tallest peak, measured at half width-half max, was found to be approximately 60 MHz. Several pictures of the mode spacing for this particular cavity length may be seen below.

For a cavity length of .329 m, the separation frequency between modes occurred 427 MHz apart. Theoretically, and calculated in the manner above, the ideal value predicted 452.8 MHz. A picture of the mode spacing for this particular cavity length may be seen below.

For a cavity length of .405 m, the separation frequency of the modes was measured to be 366 MHz with a theoretical prediction of 367.9 MHz. A picture of the mode spacing for this particular cavity length may be seen below.

For a cavity length of .515 m, the separation frequency of the modes was measured to be 273.7 MHz with a theoretical prediction of 289.3 MHz. The Doppler width of the tallest peak was observed to be 72 MHz. A picture of the mode spacing for this particular cavity length may be seen below.

For a cavity length of .523 m, the separation frequency of the modes was measured to be 268.1 MHz with a theoretical prediction of 284.9 MHz. The Doppler width of the tallest peak was observed to be 52 MHz. A picture of the mode spacing for this particular cavity length may be seen below.

A plot of the Frequency separation versus Cavity length was also performed by Carpenter and Nolen. The results of which may be viewed below.

Notice how the data fits the theoretical curve very well. This would suggest that the theory behind the Fabry-Perot laser cavity is well founded. All percent error is below 5% again reinforcing the quality work that Mr. Carpenter and Mr. Nolen performed when taking this data.

- Fowles, Grant R.
Dover Publications: New York. 1989.**Introduction to Modern Optics.** - Serway, Raymond A.
Saunders College Publishing: Philadelphia. 1996.**Physics For Scientists and Engineers with Modern Physics.**

- Abstract
- History of Wave Theory and Optics
- Michelson Interferometer
- Travelling Mirror Michelson Interferometer
- Fraunhofer Diffraction