The phenomenon of the interference of light underlies many high-precision measuring systems and displacement sensors. The use of optical fibers allows to make such devices extremely compact and economic. Two basic concepts of fiber optic interferometers are known: Mach-Zehnder and Fabry-Perot interferometers. In fiber optic interferometer Fabry-Perot the interference occurs at the partially reflecting end face surface of the fiber and an external mirror. The size of the sensitive element based on this principle can be as small as diameter of the fiber, i.e. about 0.1 mm, and the sensitivity can achieve sub-angstrom level. We can use for such an interferometer low coherence optical source (which may be even a superluminescent diode). It may be easily configured for the use in many scientific and industrial applications. Additionally, such interferometer is not sensitive to electro-magnetic interference and can be used in hostile environment. 

Let us consider the principle of operation of the fiber optic Fabry-Perot interferometer.

The radiation of the laser diode 1 is coupled into the fiber 2 and propagates through the coupler 3 to fiber 4. Then, one part of radiation is reflected from the end face of the fiber 4 and other part of radiation is flashed into the air, reflected from the mirror 5 and returned back into the fiber 4. The optical beam reflected from the end face of the fiber 4 interferes with the beam reflected from the mirror. As a result the intensity of the optical radiation at photodetector 5 is periodically  changed depending on the distance x₀ between the fiber and mirror as follows:

The displacement of the mirror by the half of the wavelength changes the path-length difference of the interfering rays by 2p, which corresponds to one period of  variation of the radiation intensity at photodetector.

On the other hand an optical radiation can not be exactly monochromatic, and consequently it has restricted coherence length. The radiation of the laser diode consists typically of several frequency modes and the total width of the spectrum Dl is equal approximately to 3-5 nm. Coherence length l_c  of such a radiation can be estimated as follows:

l_c= l²/Dl

Substituting in this equation the typical parameters of the single-mode laser diode we can find that the coherence length equals approximately 0,5 millimiter. Using the laser diode coupled with fiber Bragg grating allows the coherence length as long as many kilometers to be acheved.

The visibility (contrast) of an interference fringes depends upon the spectrum width (and, consequently, upon the coherence length) of the light. Enlargement of the path-length difference of interfering beams decreases the visibility of interference pattern. When the path-length difference reaches the coherence length, the visibility  equals  0.

The figure above shows the interference between two rays with equal intensity vs. their path-length difference l divided the coherence length l_c. This dependence is described by the equation:

where I₀  is the intensity of each of interfering beams, l is the wavelength.

Generally, the intensity of interfering rays can be essentially different (for example, in a fiber optic interferometer where the intensity of the beam reflected from the end face of the fiber about an order of magnitude less than the intensity of the radiation reflected from the mirror and returned back into the fiber). In this case 100% visibility of interference can not be achieved even at zero path-length difference of interfering rays.

where j  is the phase difference of interfering rays, I₁ and I₂ are intensities of these two rays, g is the degree of coherence.

In  fiber optic Fabry-Perot interferometer  I₁ = R₁I₀  is the intensity of the light reflected from the end face surface of the fiber and  I₂ = (1-R₁)²RI₀ is the intensity of the light reflected from an external mirror and returned back into the fiber, where  I₀ is the intensity of the laser diode radiation coupled into the fiber, R₁ is the reflectivity of the end face of the fiber and R is the reflectivity of an external mirror. For quartz fiber R₁=0,04 is Fresnel  reflectivity of the boundary surface between two substances - glass with refractive index n=1.5 and air with refractive index n=1. Thus, when the distance between interferometer mirrors equals x₀, then the light intensity detected by a photodetector is described as follows:

Generally, because of divergence of the light at the output of the fiber the percentage of radiation reflected from an external mirror and returned back into the fiber depends upon the distance between the fiber and mirror. The typical dependence of the optical power at photodetector upon the distance between the fiber and external mirror is given in the figure below.

Animation shows the computer simulation of fiber optic Fabry-Perot interferometer formed by a partially reflecting end face of the optical fiber and an external movable mirror. When the distance between the fiber and mirror is smaller than the coherence length, we can observe the interference and the intensity of the light in interferometer pulses with the mirror displacements. The visibility of interference increases with diminishing of the distance between the mirror and fiber. We can also see  in animation the image of the fiber tip reflected in the mirror. This reflection is used sometimes in practice to align the fiber perpendicularly to mirror (in this case the fiber and its reflections lie on one line that is well visible under a microscope).

Next, we shall consider the interferometric signal appearing as a result of the reflection of the light from the vibrating surface (resonator). When the resonator oscillates, the phase difference of interfering rays is varied as follows:

where l is the wavelength, x₀ is the amplitude of resonator vibration. This gives rise to the following modulation of the light intensity reflected from the interferometer cavity:

where j₀ is the phase difference between the interfering rays when the resonator is in equilibrium. Next two figures demonstrate the interferomentric signal when we change the mean separation between interferometer mirrors j₀ and amplitude of the mirror vibration x₀

Change of interferometric signal during linear increase of the mean separation between mirrors	Change of interferometric signal with linear increase of the amplitude of vibration

Expanding I(t) in a Fourier series we find the alternating components of the light modulation:

where J_i(j_w) is the Bessel functions. When j_w<<1 and j₀ = p/2+pk (k is an integer constant), then J_i(j_w) equals approximately j_w/2 and, therefore, an alternating component of intensity I(t) will be proportional to displacement of the resonator from the equilibrium: I_w~sin(wt)

And, finally, let us consider the case when the resonator is excited by external force (like oscillation of the cone in loudspeaker under the action of applied current, for example). In this case the resonator oscillation will depends on the frequency of the applied force as follows:

where Q is the quality factor of the resonator, e₀ is the resonance amplitude of oscillation and h is the frequency-dependent phase shift  between the applied force and oscillation (h varies from 0 to p, when w varies from 0 to infinity). We can see from this equation that if Q >> 1, then the amplitude of resonance oscillation is Q times bigger than the one at low frequency (or for quasi-static displacement of the resonator by the same force). Also, we can see that the amplitude of oscillation diminishes 1.414 times (square root of 2) as compared to resonance when the angular frequency of the applied force equals w_res ± w_res/2Q. So the relative width of the resonance curve equals 1/Q. In general the oscillation of resonator is superposition of several oscillations with different resonance frequencies and quality factors.