**2.4. Fiber Fabry-Perot interferometer**

As shown in **Figure 1(d)**, the fiber Fabry-Perot interferometer actually is a multiple-beam interferometer system and consists of an interferometric cavity formed by two parallel reflectors (half mirrors) with reflectance *R* on either side of an optically transparent medium with a length *lFP* [2, 4]. The available cavity structures, usually determined by the applications or design requirements, may be the extrinsic or intrinsic type [12, 20–24]. As illustrated in **Figure 1(d)**, in the extrinsic-type structure, the two reflectors are separated by an air gap or by some solid or liquid material other than the fiber, so that the measurand affects only the optical length of cavity other than the fiber. Here the fiber is used only as a light channel to transport the light beams to and from the interferometer. In contrast, in the intrinsic version, the cavity usually is constructed within the fiber as an integral part of a continuous fiber with two internal reflectors formed by flat-cut fiber ends or by fiber Bragg gratings [12, 22]. Compared with the fiber Mach-Zehnder and Michelson interferometers, the fiber Fabry-Perot interferometer is quite compact in size and therefore available as a point sensor for some applications, such as the smart structure sensing applications [23, 24].

The interference fringe, as a result of multiple reflections of the beam in the cavity, is much narrower than the normal two-beam fringes and becomes much sharper with the increase of the reflectance *R*. The finesse *F*, a parameter frequently used to characterize the overall performance of a Fabry-Perot interferometer, in relation to the sharpness of fringe, is defined as the ratio of the separation of adjacent fringes to their width FWHM (full width at half maximum), given by [4]:

$$F = \pi \sqrt{R} \,/\left(1 - R\right) \tag{5}$$

It is clear that the finesse *F* is completely decided by the reflectance *R*. Thus, for a lossless cavity, for example, we have *F* = 29.8 for *R* = 0.9 and *F* = 312.6 for *R* = 0.99. Most fiber Fabry-Perot interferometers for sensor applications, however, have low finesses which allow interferometers to operate in a linear region over a larger detection range to the measurand. If 1, the reflectance F–P and transmittance F–P of fiber Fabry-Perot interferometer can expressed, respectively, as [4]:

$$R\_{\rm F-P} \cong 2R(\mathbf{l} + \cos \phi) \tag{6}$$

$$T\_{\rm F-P} \equiv \mathbb{I} - 2R(\mathbb{I} + \cos \phi) \tag{7}$$

where φ is the round-trip propagation phase difference in the interferometer, given by

$$
\phi = 4\pi n l\_{\rm FP} \,/\,\text{\AA} \tag{8}
$$

As sensor applications, the fiber Fabry-Perot interferometer is extremely sensitive to external perturbations that affect the cavity length *lFP* as well as the refractive index *n*.

#### **2.5. Fiber Bragg grating-based Fabry-Perot interferometer**

As the wavelength selective mirrors, the fiber Bragg gratings also can be used as the intrinsic reflectors in the fiber to construct various types of fiber interferometers, such as the Michelson or Fabry-Perot interferometers.

In the physical structure, as shown in **Figure 2**, a uniform fiber Bragg grating contains a varied refractive index region with a spatial period, distributing along the fiber core within a selected length. A broadband beam propagating in the fiber will interact with each grating plane where only a part of the beam with a specific wavelength matched with the Bragg wavelength of grating will be reflected and propagates in the opposite direction, and the rest of the beam passes through this grating without obvious optical losses [9, 25]. According the mode coupling theory, the Bragg wavelength λB may be expressed as [11]:

$$\mathcal{A}\_{\mathsf{B}} = \mathcal{D} \mathfrak{m}\_{\mathrm{eff}} \Lambda \tag{9}$$

where eff denotes the effective refractive index of the fiber core; <sup>Λ</sup> is the grating period. From Eq. (9), it is evident that B only depends on eff and Λ. So B is very sensitive to ambient temperature and strain imposed on the fiber, which modify eff and Λ.

**Figure 2.** Structure and principle of a fiber Bragg grating.

*FRR* = p

F P *R R* - @ + 2 (1 cos )

F P *T R* - @- + 1 2 (1 cos )

where φ is the round-trip propagation phase difference in the interferometer, given by

4 FP

As sensor applications, the fiber Fabry-Perot interferometer is extremely sensitive to external

As the wavelength selective mirrors, the fiber Bragg gratings also can be used as the intrinsic reflectors in the fiber to construct various types of fiber interferometers, such as the Michelson

In the physical structure, as shown in **Figure 2**, a uniform fiber Bragg grating contains a varied refractive index region with a spatial period, distributing along the fiber core within a selected length. A broadband beam propagating in the fiber will interact with each grating plane where only a part of the beam with a specific wavelength matched with the Bragg wavelength of grating will be reflected and propagates in the opposite direction, and the rest of the beam passes through this grating without obvious optical losses [9, 25]. According the mode

> B eff l

where eff denotes the effective refractive index of the fiber core; <sup>Λ</sup> is the grating period. From Eq. (9), it is evident that B only depends on eff and Λ. So B is very sensitive to ambient

f =

perturbations that affect the cavity length *lFP* as well as the refractive index *n*.

**2.5. Fiber Bragg grating-based Fabry-Perot interferometer**

coupling theory, the Bragg wavelength λB may be expressed as [11]:

temperature and strain imposed on the fiber, which modify eff and Λ.

or Fabry-Perot interferometers.

f

f

respectively, as [4]:

148 Optical Interferometry

It is clear that the finesse *F* is completely decided by the reflectance *R*. Thus, for a lossless cavity, for example, we have *F* = 29.8 for *R* = 0.9 and *F* = 312.6 for *R* = 0.99. Most fiber Fabry-Perot interferometers for sensor applications, however, have low finesses which allow interferometers to operate in a linear region over a larger detection range to the measurand. If 1, the reflectance F–P and transmittance F–P of fiber Fabry-Perot interferometer can expressed,

/ 1( ) (5)

(6)

(7)

*πnl* / l (8)

= L 2*n* (9)

When two fiber Bragg gratings with identical Bragg wavelengths are imprinted in the same fiber at different positions, as shown in **Figure 1(d)**, a twin-grating-based fiber Fabry-Perot interferometer with an intrinsic cavity is constructed. If a laser beam with wavelength λ is launched into the fiber, when λ=λB, two beams partially reflected from both gratings will have a round-trip phase difference (λ) as defined in Eq. (8), where FP is an internal optical path length between two gratings. By assigning the reflectivity and transmission coefficients of each grating as (λ) and (λ), = 1, 2, respectively, the resultant reflectivity coefficient FP <sup>λ</sup> of twin-grating fiber Fabry-Perot interferometer can be proximately expressed as [26]:

$$
\eta\_{\rm FP} \left( \lambda \right) \approx \eta\_{\rm l}(\lambda) + t\_{\rm l}^2(\lambda) r\_2(\lambda) e^{-i\phi(\lambda)} \tag{10}
$$

When two gratings are of low reflectivity, we have <sup>λ</sup> <sup>=</sup> 1(λ) = 2(λ), and <sup>1</sup> <sup>λ</sup> <sup>=</sup> <sup>2</sup> <sup>λ</sup> = 1. So Eq. (10) can be rewritten as

$$\log\_{\text{FP}}\left(\lambda\right) \approx r\left(\lambda\right) \left[1 + e^{-i\phi\left(\lambda\right)}\right] = 2r\left(\lambda\right)e^{-i\phi\left(\lambda\right)/2}\cos\frac{\phi\left(\lambda\right)}{2} \tag{11}$$

The reflectance or power reflectivity FP <sup>=</sup> FP <sup>λ</sup> <sup>2</sup> of twin-grating fiber Fabry-Perot interferometer is obtained, given by

$$R\_{\rm FP}\left(\lambda\right) = \mathcal{Z}R\_{\rm B}\left(\lambda\right)R\_{\rm m}\left(\lambda\right) \tag{12}$$

where B <sup>λ</sup> <sup>=</sup> (λ) <sup>2</sup> is the reflectance of single grating, which determines the envelop of the reflection spectrum; m <sup>λ</sup> = 1 + cos(4 FP/λ) is the transfer function of the interferometer, which forms the periodically alternating spectral peaks in the reflection spectrum of the interferometer. For an input laser beam with optical power 0 and wavelength λ0, the reflected light power from the interferometer will be R <sup>=</sup> 0FP λ0 .

When the interferometer is placed in a varying environment, the reflection spectrum will shift as a whole, as a function of measurand without noticeable changes in its envelop [26]. For a Bragg wavelength shift ΔλB, which is induced by the measurand, such as temperature or strain, a new reflectance ′FP is obtained as FP′ <sup>λ</sup> = FP λ−ΔλB . When the measurand fluctuates, the reflected light power R, in turn, is altered. This feature has been well utilized in sensor applications for monitoring of the changes in the measurand through the detection of the intensity changes of the interference signal [12].

**Figure 3** displays a reflection spectrum of a twin-grating fiber Fabry-Perot interferometer with a 10-mm long intrinsic cavity, measured at 24°C. The Bragg wavelength of each grating is 1542.392 nm and the power reflectivity is about 15%.

**Figure 3.** Reflection spectrum of twin-grating fiber Fabry-Perot interferometer.
