# Principles

## Physics of Fabry-Perot Cavities

A Fabry-Perot (FP) is an optical cavity device that acts as an interferometer and thus, it is also known as a Fabry-Perot interferometer (FPI). A FPI is constructed of two optical reflectors *M _{1}* and M

_{2 }with reflectances R

_{1}and

*R*on either side of an optically transparent medium with distance of length h, as shown in Figure 1.1. The theory of FPI is reviewed in detail by Born and Wolf [1] as well as by Hernandez [2]. Charles Fabry and Alfred Perot invented FPI in 1899 [3]. Since then, FPI has been widely used for high-resolution spectroscopy. In 1981, Douglas L. Franzen and Ernest M. Kim first connected optical fibers to FP geometry [4]. The basic idea of using a single fiber to realize a fiber-optic FP cavity was presented by Cielo [5] and by Yoshino and Ohno [6]. Petuchowski et al. [7] discussed the implementation of a singlemode fiber-optic FPI (FFPI) with fiber ends to serve as mirrors. On account of the advantages of FFPI cavity, such as immunity to electromagnetic interference, capability of responding to a wide variety of parameters, very high resolution, high accuracy, and small size, it becomes an ideal transducer for many sensing applications. Some of the FFPI sensors have been successfully commercialized and widely used for health monitoring of composite materials, large civil engineering structures (e.g., bridges and dams), space aircrafts, airplanes, etc., which would lead to the realization of the so-called smart materials and structures [8].

_{2}As an interferometer, we assume transmittance *T _{i}* and reflectance

*R*1, 2, such that

_{i}, i =*R*1. We neglect the excess loss, relying on the absorption and scattering. The reflectance

_{i}+ T_{i}=*R*and transmittance

*T*are calculated using the following equations [9]:

**Figure 1.1 **Schematic of FP cavity.

where *R* is the ratio of the intensity reflected by FPI *I,* to the incident intensity *I, T* is the ratio of the transmitted intensity *I _{t}* to the incident intensity, and ф is the propagation phase shift in the interferometer, which is calculated by

where A is the optical path difference (OPD), mainly relying on the characteristics of the cavity and X is the free-space optical wavelength.

Equation 1.2 shows that *T* reaches a maximum for cos ф = —1 equal to ф = (2m + 1) n, when *m* is integral. So, the intensity of the optical spectrum changes periodically along with the optical wavelength for a certain cavity. Defining 8 *=* ф — (2m + 1)n, cos ф ~ —(1 — 5^{2}/2), then *T* approaches a maximum, with 8 ^ 1. If the reflectivities of the optical mirrors are equal and approach unity, Equation 1.2 can be simplified to

where *R = R _{1} = R_{2}* and

*T = T*= 1 — R. When 8 = 0, the transmittance reaches to the maximum. The half-high width, Аф, is defined as the phase difference between the transmittance peak and the adjacent half-maximum point. Equation 1.4 shows that, when 8 is 0,

_{1}= T_{2}*T*is the maximum and

*T*is half of its maximum value for S = ±(1 — R)A/R. So, Дф can be calculated by

The fineness *N,* a frequently used figure of merit for FP cavity, is defined as the ratio of the phase shift between adjacent transmittance peaks, 2n to Дф, which can be written as

Aiming at the filling material between the two flats of the cavity, the FFPI sensors are generally classified into intrinsic and extrinsic ones. No matter what kind the sensor is, the fiber plays an important role in FFPI. Light emitted from the light source propagates along the fiber to the interference structure. Then, the detected signal couples to the photodetector along the fiber. The differences between intrinsic and extrinsic FFPI sensors mainly focus at the filling material of the cavity. For intrinsic FFPI sensors, the cavity formed by two separated mirrors is filled by the fiber itself, as shown in Figure 1.2a. But for extrinsic FFPI sensors, the filling of the cavity can be an air gap or some other materials except for the fiber, as shown in Figure 1.2b.

However, in this book, we will classify the FFPI sensors in another way, by focusing on the quantity of the interference beams. In the following sections of this chapter, we will introduce three major types of FFPI.