Interferometric Fiber-Optic Sensors PDF

OPTICAL FIBER SENSORS Advanced Techniques and Applications EDITED BY Ginu Rajan  3 Interferometric Fiber-Optic Sensors Sara Tofighi, Abolfazl Bahrampour, Nafiseh Pishbin, and Ali Reza Bahrampour CONTENTS 3.1 Optical F...

OPTICAL FIBER SENSORS Advanced Techniques and Applications EDITED BY Ginu Rajan  3 Interferometric Fiber-Optic Sensors Sara Tofighi, Abolfazl Bahrampour, Nafiseh Pishbin, and Ali Reza Bahrampour CONTENTS 3.1 Optical Fibers and Their Characteristics................................................................................. 38 3.1.1 Standard Optical Fibers............................................................................................... 38 3.1.2 Photonic Crystal Fibers............................................................................................... 38 3.1.3 Quasi-Photonic Crystal Fibers..................................................................................... 39 3.1.4 Polarization-Maintained Optical Fibers......................................................................40 3.1.5 Slab Optical Waveguides.............................................................................................40 3.2 Electromagnetic Interference................................................................................................... 41 3.2.1 Field Interference and First-Order Correlation Function............................................ 41 3.2.2 Second- and Higher-Order Interferences..................................................................... 42 3.2.3 Quantum Theory of Correlation Functions................................................................. 43 3.3 Multiport Optical Fiber............................................................................................................44 3.3.1 Beam Splitter...............................................................................................................44 3.3.2 Multiport Optical Fiber Beam Splitter........................................................................ 45 3.3.3 Unitary Transformation as a Multiport Optical Fiber Beam Splitter.......................... 45 3.3.4 Matrix Theory of Multiport Optical Fibers.................................................................46 3.4 Multiport Optical Fiber Interferometers.................................................................................. 47 3.4.1 Fabry–Pérot Interferometer......................................................................................... 47 3.4.2 Fiber Bragg Gratings................................................................................................... 49 3.4.3 Sagnac Interferometer.................................................................................................. 50 3.4.4 Fiber Ring Resonator Interferometer........................................................................... 50 3.4.5 Mach–Zehnder Optical Fiber Interferometer.............................................................. 51 3.4.6 Michelson Optical Fiber Interferometer...................................................................... 53 3.4.7 Modal Optical Fiber Interferometer............................................................................54 3.4.8 Moiré Optical Fiber Interferometer............................................................................. 55 3.4.9 White Light Optical Fiber Interferometer................................................................... 55 3.4.10 Composite Optical Fiber Interferometer...................................................................... 57 3.5 Signal Recovering and Noise Source in Optical Fiber Interferometry................................... 58 3.5.1 Signal Recovery Method............................................................................................. 58 3.5.2 Phase-Generated Carrier Homodyne Detection.......................................................... 58 3.5.3 Fringe-Rate Method..................................................................................................... 58 3.5.4 Homodyne Method...................................................................................................... 59 3.5.5 Noise Sources in OFIs................................................................................................. 59 3.6 Interferometric Optical Fiber Sensors (IOFSs)........................................................................60 3.6.1 Principle of Operation of IOFSs..................................................................................60 3.6.2 Principle of Plasmon.................................................................................................... 61 3.6.3 Plasmon Cooperated IOFS.......................................................................................... 63 3.6.4 Fabrication Methods of IOFS...................................................................................... 63 37 38 Optical Fiber Sensors: Advanced Techniques and Applications 3.7 Applications.............................................................................................................................64 3.7.1 Temperature Measurement–Based Sensors.................................................................64 3.7.2 Strain Measurement–Based Sensors...........................................................................66 3.7.3 Refractive Index Measurement–Based Sensors...........................................................68 3.7.4 Quantum Mechanical Applications............................................................................. 69 References......................................................................................................................................... 72 3.1 OPTICAL FIBERS AND THEIR CHARACTERISTICS Fiber-optic lines have revolutionized long-distance phone calls and Internet. Optical fibers are widely used for communications and fiber sensors, thanks to their high speed, large bandwidth, reliability, and low attenuation. They are mainly made of silica or polymers. Depending on the fiber applica- tions, different structures such as conventional fiber, photonic crystal fiber, and quasi-photonic crys- tal fibers are designed. A brief review of different fiber structures is presented in this section. 3.1.1 Standard Optical Fibers On the basis of total internal reflection, light is guided by standard optical fibers. A standard optical fiber consists of two different coaxial cylindrical layers. Core is the central region, which is sur- rounded by a cladding layer. The core’s refractive index is greater than the cladding’s. Most of the energy of the guided modes propagates in the core and a small fraction of the total energy propagates in the cladding region. The cladding radius compared to the core radius is so large that the surround- ing medium has no effect on the light propagating inside the optical fiber. Depending on the appli- cation, optical fibers are made from glass, polymers, or crystals. Conventional or communication optical fibers are made of silica (SiO2) glass; the core and cladding refractive indices are adjusted by employing suitable dopants such as GeO2, Al2O3, and B2O3 impurities. The optical fibers are charac- terized by the core–cladding index difference Δ and normalized frequency v parameters. Δ and v are defined by the following equations Δ = (n1−nc)/n1 and v = k0 a n12 − nc 2 , where n1 and nc are the core and cladding refractive indices, respectively, k0 = 2π/λ is the wave number, a is the core radius, and λ is the light wavelength. The number of modes supported by the optical fiber is determined by v. Optical fibers designed for v < 2.405 support single-mode operation. The Δ parameter is about 0.003 for a single-mode fiber. The main difference between the single- and multimode fiber is the core size. Single-mode fibers require a core size of a < 5 μm. An outer radius of b = 62.5 μm is commonly used for both single- and multimode fibers. Optical fibers with low Δ parameters are named weakly guiding fibers. The weakly guiding modes are denoted by LPμν, where μ and ν are integer numbers. The fundamental mode of optical fiber (HP11) corresponds to the LP01 weakly guiding mode. The normalized propagation constant β versus the dimensionless frequency is called the dispersion curve. Obviously, the frequency dependence of group velocity vg = (dβ/dω)−1 leads to pulse broaden- ing because different spectral components of the pulse do not arrive simultaneously at the fiber output. The dispersion parameter D = d/dλ(1/vg) is expressed in units of Ps/(Km ⋅ nm). D consists of material dispersion, waveguide dispersion, polarization mode dispersion, and mode dispersion. In single-mode fibers, the mode dispersion parameter vanishes. Fiber loss is another limiting fac- tor that reduces the fiber output power and it depends on the atomic absorption and scattering. Depending on the optical fiber parameters, light wavelength, and coupling conditions; bounded, radiation, and evanescent modes can be excited. However, evanescent modes cannot propagate along the optical fiber and store energy near the excitation source. 3.1.2 Photonic Crystal Fibers Photonic crystal fibers (PCFs) refer to another class of optical fibers that have wavelength-scale mor- phological microstructures running down their length. PCFs can be divided into different categories Interferometric Fiber-Optic Sensors 39 (a) (b) (c) FIGURE 3.1 Schematic cross section of (a) HC-PCF, (b) solid-core PCF (holey fiber), and (c) Bragg fiber. based on their guidance mechanism or photonic crystal dimensionality in their transverse plane. Based on their guiding mechanisms, they are divided into index-guiding PCFs (IG-PCFs) and photonic band- gap fibers (PBFs). Over the last decade, both types of PCFs have been studied, but particular attention has been given to PBFs due to their lattice-assisted light propagation within a hollow core. This par- ticular feature indeed has a number of advantages such as lower Rayleigh scattering, reduced nonlinear- ity, novel dispersion characteristics, and potentially lower loss compared to conventional optical fibers. In addition, the hollow-core PCFs (HC-PCFs) also enable enhanced light–material interaction, thus providing a valuable technological platform for ultrasensitive and distributed biochemical sensors. These different types of PCFs are shown in Figure 3.1. As mentioned before, light guidance in HC-PCF (Figure 3.1a) is based on the bandgap effect in photonic crystals. For a given value of propagation constant (β), if the transverse component of the wave vector falls into the bandgap region, no transverse propagation is allowed. So the propagation mode is confined to the hollow defected core of the PCF. It is expected that the attenuation of the hollow PCF is much smaller than a standard optical fiber, but due to the surface capillary wave production during the PCF fabrication, its attenuation is about 1 dB/km. The guidance of light in solid-core PCFs (Figure 3.1b) is due to the modified total internal reflec- tion. The average refractive index of the cladding in a solid-core PCF is lower than the core’s refrac- tive index. In fact, here, the average refractive index is not the geometric average, but an effective refractive index (neff) corresponding to the largest possible value of the photonic crystal propagation constant (βmax(ω)) at a given frequency. In the effective refractive index method, neff attributed to the cladding of solid-core PCFs is equal to the modal index of fundamental space-filling model ( neff = βmax (ω)c /ω) [10,11]. Light at a given frequency and a β greater than βmax cannot propagate in the photonic crystal structure; hence, modified internal reflection is a specific case of bandgap guidance. Some authors also use the term microstructured optical fiber (MOP) for referring to PCFs where guidance results from bandgap effect. Bragg fibers are a special case of PCFs. A Bragg fiber is a concentric arrangement of dielectric layers wrapped around a core, which may or may not be hollow. The periodicity of the photonic crystal cladding in Bragg fiber is 1D. Light propagation in the Bragg fiber (Figure 3.1c) is based on Bragg diffraction effect. Bragg fibers with full circular symmetry and without birefringence can be single-mode and single-polarization fibers. 3.1.3 Quasi-Photonic Crystal Fibers The geometry of the cladding (microstructured) plays an important role in the bandgap regime, but all HC-PCF fibers are based on periodic structures and thus are limited to a few geometries such as a triangular, square, honeycomb, or Kagome periodic array lattice. However, recently, few works showed that it is possible to explore new functionalities and increase the degree of freedom 40 Optical Fiber Sensors: Advanced Techniques and Applications (a) (b) FIGURE 3.2 (a) 12-fold PQ fiber and (b) modified 8-fold. of HC-PCFs by studying new kinds of fibers based on photonic quasi-crystals (PQs). We define PQs as aperiodic photonic crystal (1D, 2D, or 3D) that lack translational symmetry and instead are extremely rich in rotational symmetry. PQ structures have presented intriguing achievements in optics and crystallography. Optical properties such as complete photonic bandgap, laser micro- cavity, and guided resonance in different aperiodic structures have been investigated and reported [13–16]. An opulent rotational symmetry provides a bandgap in the lower refractive index region, which can lead to HC fibers with minimum air–glass refractive index contrast. Recently, PQs have been investigated principally for index-guiding mechanism and Sun et al. for the first time proposed an HC PQ fiber based on a 12-fold symmetric structure as shown in Figure 3.2a. They demonstrated that 12-fold fibers exhibit a double photonic bandgap. Different structures such as a 12-fold and a modified 8-fold symmetric structure (Figure 3.2b) have been proposed and simulated where a double bandgap with λ/Λ < 1 (λ is the wavelength and Λ is a defined pitch) has been demonstrated. Other simulations on PQ fibers confirm an extremely lower loss (nominally). 3.1.4 Polarization-Maintained Optical Fibers Birefringent optical fibers are fibers with anisotropic cross sections that have two distinct principle axes with different refractive indices, called the fast and slow axis. These are named so because there exist two different light velocities that depend on the polarization alignment of the incident light with one of the principle axes. For a light beam whose polarization is aligned with one of the principle axes of the birefringent fiber, the light’s polarization is kept constant during propagation along that optical fiber. The birefringence parameter of the fiber is defined by the difference between the two refractive indices corresponding to the two principle axes B = ns−nf , where ns and nf are the refrac- tive indices of the slow and fast axis, respectively. The fiber beat length LB = λ/B is defined as the fiber length over which the phase difference between the fast and slow waves becomes 2π radians. To preserve the polarization direction, perturbation periods introduced in the drawing process as well as the physical bend and twists must be greater than the beat length. This kind of optical fibers are called polarization-maintained fibers (PMF). PMFs are employed in different interferometric ­optical fiber sensors (IOFS) such as optical fiber gyroscopes and intruder detection systems. PCFs with asymmetric microstructured in either cladding or the core region could exhibit strong birefrin- gent compared to the conventional PMFs. Figure 3.3 shows the cross section of high-birefringence photonic crystal fiber (HiBi-PCF) and polarization-maintaining photonic crystal fiber (PM-PCF). 3.1.5 Slab Optical Waveguides Optical fibers are suitable for long-length sensors such as oil pipeline leak detection systems and intruder sensors. In many applications such as integrated circuits and miniaturized sensors, the transmission length is less than a few millimeters, so slab optical waveguides are used for these Interferometric Fiber-Optic Sensors 41 (a) (b) FIGURE 3.3 Schematic cross section of (a) HiBi-PCF and (b) PM-PCF. short-length applications. A dielectric waveguide consists of a dielectric with refractive index n1, which is deposited on a lower refractive index substrate; the refractive index of the surrounding medium is also smaller than n1. The modes propagating in the slab waveguides are transverse electric (TE) and transverse magnetic (TM) modes. The mathematical model analysis of slab waveguides can be found in any standard text book. The narrow dielectric strip waveguide modes are also clas- sified into bounded, radiation, and evanescent modes [4,20]. The slab optical waveguides can also be employed in quantum computing unitary gates. 3.2 ELECTROMAGNETIC INTERFERENCE Interference is a wave effect that can be observed in any type of wave whether it be electromagnetic, acoustic, elastic, or matter waves. Generally, field or first-order interference is just called interfer- ence, while higher-order interferences can be employed for measurement of the electromagnetic wave characteristics such as statistical parameters. The interference behavior is formulated by the correlation function. 3.2.1 Field Interference and First-Order Correlation Function Most of the well-known interferometers such as Fabry–Pérot, Sagnac, Michelson, and Young ­double-slit interferometers are operating on the basis of field interference. The intensity correspond- ing to the field E(x1, t1, x2, t2) = αE(x1, t1) + βE(x2, t2) results in interference fringes. As an example, in Young’s double-slit experiment, E(x1, t1) and E(x2, t2) are the fields on the screen corresponding to the slits at positions (x1, t1) and (x2, t2), respectively. The intensity at the observation point (x, t) is given by I ( x, t ) = †α I ( x1, t1 ) + β I ( x2 , t2 ) + 2 †Re αβ*E ( x1, t1 ) E* ( x2 , t2 ) ( ) 2 2 (3.1) The third term on the right-hand side of Equation 3.1 corresponds to the field interference. In con- ventional interferometer, light can propagate from (x1, t1) and (x2, t2) points, through an arbitrary medium to the observation point (x, t). In optical fiber interferometers (OFIs), the transformation media may be one or more optical fibers. The first-order correlation function G (1)(x, t; x′, t′) is defined by 1 G ( ) ( x, t; xʹ, t ʹ ) = †E ( ) ( x, t ) E ( ) ( xʹ, t ʹ ) − + (3.2) where E(+)(x) and E(−)(x) are the positive and negative frequency parts of the electric field 〈…〉 stands for the ensemble average 42 Optical Fiber Sensors: Advanced Techniques and Applications For a stationary field, the statistical description is invariant under time-variable displacements ( 1 1 ) G ( ) ( t , t ʹ ) = G ( ) ( t − t ʹ = τ ). The random classical fields are usually stationary and have the ergodic property; therefore, the ensemble average has the same value as the time-averaged correlation func- tion  (1) (t − t ʹ ) [22–24]: T 1 1 G (1) ( x, xʹ, τ ) =  ( ) ( x, xʹ, τ ) = lim ( x, t1 + τ ) E ( +) ( xʹ, t1 ) dt1 ∫E (−) (3.3) T →∞ T 0 No fringes will be observed if the correlation function G (1)(x, x′) vanishes and it can be considered that the fields at x and x′ are incoherent. On the other hand, the highest degree of coherence is asso- ciated with a field that exhibits the strongest possible interference fringe. The strength of interfer- ence is defined by the visibility factor as follows: 1 1 I −I 2 G ( ) ( x, x )G ( ) ( xʹ, xʹ) V =  max min =  (1) (3.4) I max + I min 1 G ( x, x ) + G ( ) ( xʹ, xʹ) If the fields incident on the two pinholes have equal intensity, the visibility varies between V = 0 for incoherent light and V = 1 for first-order coherent light. Generally, the first-order normalized correlation function is 1 1 G ( ) ( x, xʹ) g ( ) ( x, xʹ) = † (3.5) 1 1 G ( ) ( x, x )G ( ) ( xʹ, xʹ) 1 The necessary condition for coherence is g ( ) ( x, xʹ ) = 1 that is equivalent to the factorization ( ) property of correlation function G (1) ( x, xʹ ) = F ( x )F ( xʹ) and normalized correlation function ( 1 ) g ( ) ( x, xʹ ) = f ( x ) f ( xʹ) [22,23]. 3.2.2 Second- and Higher-Order Interferences In second-order interferometry, the intensities at two different positions are measured, multiplied, and averaged. In Hanbury Brown and Twiss (HB-T) interferometer, the intensities at two different positions ri (i = 1, 2) are detected individually. The detectors’ outputs, which are in the low-frequency range, are transmitted to a central correlating device where they are multiplied and the product is averaged: 2 E ( ) (ri , t ) = Ek + Ek ʹ + Ek Ek*ʹe ( ) i + Ek*Ek ʹe ( ) i + 2 2 i k − k ʹ ⋅r − i k − k ʹ ⋅r (3.6) where Ek, Ek′, k, and k′ are the amplitudes and wave vectors of the incoming fields, respectively. The advantage of HB-T method was to detect the signals first, then filter the high-frequency com- ponents so that the signals are of relatively low frequency and can transmit undisturbed over large distances. This method deals with signal intensities and is quite different from the field interferometric method that works with the average of the product of two random fields. Hence, the inputs of HB-T experiment have no rapid oscillations. The average of the product of the two intensities from (3.6) is 2 2 2 E ( ) (r1, t ) †E ( ) (r2 , t ) + + = (E k 2 + Ek ʹ 2 ) + 2 Ek Ek ʹ 2 2 cos[( k − kʹ ) ⋅ (r1 − r2 )] (3.7) Interferometric Fiber-Optic Sensors 43 Clearly, the third term on the right-hand side of Equation 3.7 represents an interference effect. The second-order correlation function G(2)(r1, t1; r 2, t2) is defined as 2 G ( ) ( r1, t1; r2 , t 2 ) = E ( ) ( r1, t1 ) E ( ) ( r2 , t 2 ) E ( ) ( r2 , t 2 ) E ( ) ( r1,tt1 ) − − + + (3.8) G(2)(r1, t1; r 2, t2) is defined by ensemble averages rather than the time averages. That is, it is assumed that the incoming wave is a stochastic electromagnetic radiation and has the ergodic property. As shown in Equation 3.8, G(2)(r1, t1; r 2, t2) is a measure of the strength of intensity interference. The normalized second-order correlation function is defined by 2 ( 2) G ( ) ( x1, x2 ) g ( x1, x2 ) = 1 1 (3.9) G ( ) ( x1, x1 ) G ( ) ( x2 , x2 ) 2 The second-order coherent light is defined by g ( ) ( x1, x2 ) = 1, which is equivalent to the factoriza- tion property of the second-order correlation function. The first- and second-order coherency can be found in radiation generated by natural sources. In general, man-made sources such as lasers and radio transmitters can have much higher regularity than is ever possible for natural sources. In order to better understand the concept of coherence, higher-order correlation functions are defined by G ( ) ( x1,…, xn ; xn ,…, x1 ) = E ( ) ( x1 )… E ( ) ( xn ) E ( ) ( xn )… E ( ) ( x1 ) n − − + + (3.10) The nth-order normalized correlation function is G ( ) ( x1,…, xn ; xn ,…, x1 ) n g ( ) ( x1,…, xn ) = n n 1/ 2 (3.11) ∏ { 1 G( ) ( x j , x j ) j =1 } The Mth order coherence field is defined by g ( ) ( x1,, x2 n ) = 1 for all n ≤ M and all combinations of n arguments of xi. Full coherence requires coherency for all orders of correlation function. 3.2.3 Quantum Theory of Correlation Functions The most popular quantum interference arrangement is the Hong–Ou–Mandel interferometer [26,27]. As shown in Figure 3.4, in Hong–Ou–Mandel interferometer, a pair of photons enters a beam splitter (BS) that can be a lossless optical fiber coupler (OFC) or a conventional BS. In general, there are four possible outputs for the BS: (a) Both are reflected. (b) Both are transmitted. (c and d) One is transmitted, while the other is reflected. CC Detector BS (a) (b) (c) (d) FIGURE 3.4 4 Four possible outputs of BS in Hong–Ou–Mandel interferometer: (a) both photons are reflected, (b) both photons are transmitted, (c and d) one photon is transmitted while the other is reflected. CC and BS stand for coincident count and beam splitter, respectively. 44 Optical Fiber Sensors: Advanced Techniques and Applications Here, (c) and (d) are degenerate states. Due to energy conservation in a symmetric BS, there is an overall π-phase difference in the photons of the first two states (a) and (b) that leads to destructive interference. Hence, the first two cases completely cancel each other out. If the input state is ∙1,1〉, then the output state will be | Ψ out = 1/ 2 | 2, 0 − | 0, 2. ( ) Due to the destructive interference described earlier, the ∙1,1〉 term disappears at the output of the ( ) BS | Ψ out. The Bosonic property of the photons dictates that the photons will have the tendency to go together to either side of the BS. Generally, for a quantum system which is characterized by a matrix density (ρ), and an expecta- tion value of an arbitrary observable (O), which is given by O = Tr{ρO}. The average counting rate of an ideal photodetector which operates based on single-photon absorption is proportional to the expectation value of the observable E(−)(x)E( + )(x) (Tr{ρE(−) (x)E( + )(x)}), where E( + ) (x) and E(−)(x) are the positive and negative frequency part of the electric field operator Tr{ρE(−) (x)E( + )(x)}. Similarly, the n-photon absorption rate is obtained by nth-order quantum correlation function as { − + + } G ( n ) ( x1,…, xn ; xn ,…, x1 ) = Tr ρE ( ) ( x1 )… E ( ) ( xn ) E ( ) ( xn )… E ( ) ( x1 ) − (3.12) If there is an upper bound (M) on the number of photons present in the field, then the function G(n) vanishes for all orders higher than that fixed M. 3.3 MULTIPORT OPTICAL FIBER Linear multiport OFCs are a generalization of the BS. The BS has an essential role in many experi- ments of classical and quantum optics. The simplest BS can be considered as a four-port device. An optical fiber BS (OFBS) or OFC is a pair of coupled optical fibers with their cores brought alongside one another. The evanescent field of each fiber core excites the mode of the other fiber and these modes are then coupled. OFBSs are fabricated by placing a pair of single-mode fibers side by side, twisting together and fusing while elongating the contact region. In order to better understand the operation of multiport OFCs, a simple BS will be studied first in the following section. 3.3.1 Beam Splitter A BS can be described by two input and two output modes of the radiation field. The operation of a lossless BS can be described by a unitary scattering matrix. In a lossless BS, the output fields (Eout) are related to the input fields (Ein) through a 2 × 2 unitary scattering matrix (): ⎛ E1out ⎞ ⎛ sin Ω eiϕ cos Ω ⎞ ⎛ E1in ⎞ (3.13) ⎜ ⎟=⎜ ⎟⎜ ⎟ ⎝ E2 out ⎠ ⎝ cos Ω −eiϕ sin Ω ⎠ ⎝ E2in ⎠ φ is the phase difference between the input fields, which can be generated by placing an external phase shifter (PS) placed before the BS in one of the input ports. The parameter Ω is related to the reflectivity and transmittance of the BSs via R = cos2 Ω and T = sin2 Ω, respectively [28,29]. The transfer matrix of an optical fiber BS  can be described by a coupling constant κ and a coupling length L. The coupling coefficient κ is determined by the wavelength of light and is an exponentially decaying function of interfiber distance: ⎛ cosκL isinκL ⎞  =⎜ ⎟ (3.14) ⎝ isinκL cosκL ⎠ For κL = π/4, a symmetric 2 × 2 coupler (or 3 dB coupler) is obtained [30,31]. Interferometric Fiber-Optic Sensors 45 PS OFC OFC OFC 1΄ OFC N 2΄ N–1 N–2 OFC OFC (N – 1)΄ 1 N΄ FIGURE 3.5 A schematic diagram of multiport OFBS built of 2 × 2 OFCs and PSs. 3.3.2 Multiport Optical Fiber Beam Splitter A generalized BS can be built from commercial optical components. As shown in Figure 3.5, the implementation of a generalized N port OFBS can be achieved by a suitable configuration of 2 × 2 OFBSs and PSs. For tritters, PSs φ1 = tan(1/3) and φ2 = φ1/2 are needed, while for quarter, a PS of φ = π/2 is required. A multiport OFBS consists of N coupled optical fibers. N optical fibers are placed side by side, twisting and fusing them together while elongating the contact region. A generalized OFBS is a 2N-port optical fiber network. Due to the energy conservation law, any 2N-port optical fiber network can be described by an N × N unitary matrix. The matrix elements of a generalized OFBS are wave- length dependent and can be determined versus the distances between the fiber cores. The matrix element  ij is the ratio of the electric field at port j to the electric field at port i, while there is no input light at the other ports. In many cases, the forward and backward modes can be considered as the input and output of the multiport interferometer, respectively. 3.3.3 Unitary Transformation as a Multiport Optical Fiber Beam Splitter As mentioned in the previous section, due to the energy conservation law, any lossless multiport OFBS can be described by a unitary matrix. Moreover, it was shown that for any unitary matrix, there exists an experimental setup consisting of PSs and simple 2 × 2 BSs [29,32]. The multiport BS can be viewed as a black box transforming N external inputs into N external outputs. The internal inputs (outputs) are those that are internally connected to the output (inputs) of another BS inside the system. The transfor- mation matrix of a multiport BS can be obtained by employing the scattering matrices of the 2 × 2 BSs and PSs. After eliminating the internal inputs and outputs of 2 × 2 BSs, the external input–output matrix of multiport BS, which consists of four submatrices, is obtained as follows:  ( ext )  ⎛ Eout ⎞ ⎛ See Sei ⎞ ⎛ Ein( ext ) ⎞ (3.15) ⎜  ⎟=⎜ ⎟⎜  ⎟ ⎜ E (int ) ⎟ ⎝ Sie Sii ⎠ ⎜⎝ Ei(nint ) ⎟⎠ ⎝ out ⎠ For example, the submatrix Sei describes the transformation from the internal inputs of the BS to the external outputs. The connection between BSs is described by a connection matrix . The matrix elements ij are the phase shifts internally accumulated by fields evolving between the BSs:  ( int )  Eout =  Ein(int ) (3.16) 46 Optical Fiber Sensors: Advanced Techniques and Applications The input–output relation of the whole system can be obtained by solving Equations 3.15 and 3.16 simultaneously:  ( ext )  Eout ( −1 ) = See + Sei (  − Sii ) Sie Ein( ext ) (3.17) The unitary matrix of a complex multiport BS involving many BSs and PSs is simply obtained by −1  = See + Sei (  − Sii ) Sie (3.18) Symmetric multiports are a special class of unitary multiports. All the matrix elements are of the same modulus. A field at any of the inputs is a coherent superposition of all output modes with equal modulus of the amplitude. If intensity I enters one input of a symmetric N × N multiport, the intensity at any output is I / N. The generic form of the transfer matrix of a symmetric N × N multiport can be expressed versus the roots of unity zN = exp(i(2π/N )) as 1 ( m −1)( k −1)  (mkN) = zN (3.19) N As an example, the transfer matrix of 3 × 3 symmetric multiport (which is called a tritter) is as follows: ⎛1 1 1⎞ 1 ⎜ ⎟  ( 3) = 1 α α2 ⎟ (3.20) 3 ⎜⎜ ⎝1 α2 α ⎟⎠ where α ≡ ei2π/3. Symmetric multiports that can be transformed into one another by simple renum- bering of inputs and outputs or by including PSs at the inputs and outputs can be considered as an equivalent class. Each of the 2 × 2 BSs and 3 × 3 symmetric multiports has only one equivalence class. 3.3.4 Matrix Theory of Multiport Optical Fibers In the previous sections, it was assumed that light was linearly polarized in all parts of the multiport and there was no element in the optical fiber network to change the polarization. In this section, the Jones vectors and matrices [24,33] are employed to introduce a method for analyzing complex interferometric fiber systems, including polarization and back reflection effects. The polarization state of quasi-plane wave-front is described by a Jones vector in a complex 2D ( ˆ iδ E0eiωt − ikz , where ∙a∙2 + ∙b∙2 = 1 and δ is the phase difference between vector space E = aî + bje ) the x and y components of the electric field. Any common phase in a and b can be taken out to be absorbed by the phase term eiωt − ikz. Every linear optical element in each branch of the interferometer is represented by a 2 × 2 matrix. Systems with bidirectional propagation may be faithfully simulated with scattering matrices, that is, bidirectional Jones matrices can be generalized to an arbitrary OFI. A two-port device is described Interferometric Fiber-Optic Sensors 47 by a 4 × 4 scattering matrix, while a four-port device such as fiber-optic coupler is presented by 8 × 8 scattering matrices. In general, a bidirectional vectorial 2N-port OFI can be described by a 4N × 4N scattering matrix. The forward and backward electric field vectors are denoted T T by E f = E (f1) , E (f2 ) ,, E (f2 N ) and Eb = Eb(1) , Eb( 2 ) ,, Eb( 2 N ) , respectively, where E (fi ) = E (fxi) , E (fyi) ( ) ( ) ( ) ( and Eb(i ) = Ebx (i ) , Eby) (i ). The forward and backward fields are related through the scattering matrix S(Eb = SEf) where the matrix elements Sij are 2 × 2 block matrices given by the backward field at port i when the forward fields at other ports except port j are zero Eb(i ) = S ij E (f j ). The 2 × 2 block matrices Sij are quasi-Jones matrices, and their elements Sklij (k, l = x, y) are the ratio of Ebl(i ) /E (fkj ) when all other field components E fm (q) (q ↑ j,†m ↑ k ) are zeros. 3.4 MULTIPORT OPTICAL FIBER INTERFEROMETERS Interferometry is based on superimposing two or more light beams to measure the phase difference between them. These beams have the same frequency. In classical experiments and interferometric sensors, all the light beams are generated by a given light source, even though from the quantum point of view the interference of the beams from different sources is of great importance. Typically, an incident light beam in an interferometer is split into two or more parts and then recombined together to create the interference pattern. To consider the interference fringes, at least two optical paths are necessary for an interferometry experiment. These paths can be in a single- or multimode optical fiber. In OFIs utilizing multimode optical fibers such as the Sagnac interferometer, each mode defines an individual optical path. In Sagnac interferometer, the optical paths are defined by the clockwise (CW) and counterclockwise (CCW) optical fiber modes. The optical path can be defined by separate single-mode optical fibers such as in Mach–Zehnder OFI. The maximum and minimum points of the fringes correspond to even and odd numbers of half-wavelength optical path differences (OPDs), respectively. There are many interferometer configurations that have been realized with the optical fiber. Some configurations such as Fabry–Pérot, fiber Bragg gratings (FBGs), Sagnac bire- fringence OFI, Mach–Zehnder, Michelson, and Moiré interferometer are presented in this section. 3.4.1 Fabry–Pérot Interferometer The simplest OFI is the Fabry–Pérot interferometer (FPI). It consists of two parallel reflectors with reflection coefficient R1(ω) and R2(ω) separated by a cavity length L. These reflectors can be mirrors, interface of two dielectrics, or FBGs. The cavity may be an optical fiber or any other medium. The FP reflectance RFP and transmittance TFP versus the mirror’s reflection coefficient are obtained by R1 + R2 + 2 R1R2 cos φ RFP = (3.21) 1 + R1R2 + 2 R1R2 cos φ T1T2 TFP = (3.22) 1 + R1R2 + 2 R1R2 cos φ where ϕ = 4πnL/λ is the round-trip propagation phase shift in the interferometer n is the refractive index between the reflectors λ is the free-space optical wavelength The FP transmittance has its maximum at the resonance frequencies corresponding to the round- trip propagation phase ϕm = (2m + 1)π, where m is an integer number. The detuning phase is defined 48 Optical Fiber Sensors: Advanced Techniques and Applications by Δ = ϕ−ϕm. For high reflectance mirrors, the transmission coefficient near the resonance fre- quencies can be written as follows: T2 TFP = 2 (3.23) (1 − R ) + RΔ 2 where R = R1 = R2, T = 1−R, and δ = ± (1 − R) / R is the phase corresponding to the FP bandwidth. The FP finesse can be written as π R F = (3.24) 1− R In an interferometer with lossless mirrors R = R1 = R2 = 0.99, the finesse is equal to F = 312.6, which is considered very high. For the mirrors’ reflectance R = R1 = R2 ≪ 1, the FP reflectance and trans- mittance can be approximated by RFP ≅ 2 R(1 + cos φ) (3.25) TFP ≅ 1 − 2 R(1 + cos φ) (3.26) The concept of finesses is not suitable for R ≪ 1 FPs. The finesse equals one for R = 0.172 and is undefined for R < 0.172. Due to the energy conservation law, the reflection and transmission coefficients of FP follow R + T = 1 equation. This equation made it possible to measure the FPI resonance frequency by employing the FP transmission or reflection coefficient. If in the optical fiber FPI the mirrors are separated by a single-mode optical fiber, then the opti- cal fiber FPI is called intrinsic fiber FP interferometer (IFFPI). However, in the extrinsic fiber FP interferometer (EFFPI), the two mirrors are separated by an air gap or some material other than fiber. Light from emitter to the FP and from FP to the detector is generally transmitted by single- mode fibers. Three schematic configurations of IFFPI are shown in Figure 3.6. As shown in Figure 3.6a, one end of the fiber is polished as a mirror. For higher-quality factor (Q = ω/Δω), the polished end is coated with suitable dielectric layers. The second mirror of IEFPI is an internal mirror that can be made by splicing polished fibers or polished coated fibers. The mirrors of an IFFPI presented in Figure 3.6b are internal fiber mirrors, while those used in the IFFPI shown in Figure 3.6c are FBG reflectors. Depending on the application of IFFPI, one of the presented configurations can be used. Four different configurations for EFFPI are also shown in Figure 3.6. Figure 3.6d shows an EFFPI with air gap cavity bounded by the ends of a polished fiber and a diaphragm mirror. The cav- ity length is of the order of several microns and can be increased by convex mirror diaphragm. To increase the EFFPI quality factor as shown in Figure 3.6e, a thin film of transparent solid material is also coated on the end of the optical fiber. The EFFPIs of Figure 3.6d and e are operating based on the reflection coefficient measurement. The air gap cavity between two polished fiber surfaces where the fibers are aligned in a hollow tube is another EFFPI configuration (Figure 3.6f). The structure of the in-line fiber etalon is shown in Figure 3.6g. This structure is an HC fiber spliced between two single-mode fibers. The diffraction loss imposes a limit of the order of a few hundreds of microns on the EFFPI’s practical length. Interferometric Fiber-Optic Sensors 49 Internal mirror Polished end Internal mirrors (a) IFFPI FBG (b) (c) Internal Diaphragm mirror Capillary tube mirror (f ) (d) EFFPI Transparent Internal media External Hollow-core Single-mode mirror mirror fiber fiber (e) (g) FIGURE 3.6 Different configurations of (a–c) IFFPI and (d–g) EFFPI. 3.4.2 Fiber Bragg Gratings The forward and backward modes of optical fibers propagate without coupling to each other in the absence of any perturbation. Mode coupling can be controlled by changing the refractive index along the optical fiber. The coupled mode theory (CMT) is a standard subject in related text books that explain mode propagation in slightly nonuniform media. In a periodic structure, where the refractive index of the optical fiber core varies periodically from a lower index n 0 to a higher index n, the scattering from different periods can produce constructive interference for some fre- quencies and destructive interference for other frequencies in the forward and backward modes. This well-known effect is called Bragg diffraction. Depending on the period length (Λ), periodic structures are classified as long-period grating (LPG) or FBG. The periods of the LPG and FBG are of the order of microns and nanometers, respectively. The LPG operation is based on the coupling of fundamental core modes to higher-order copropa- gating cladding modes. The coupling wavelength is obtained by the phase matching condition or linear momentum conservation law λ = (β1 − β2)Λ, where β1 and β2 are propagation constants of the core and cladding modes, respectively. FBGs are employed as a frequency-selective mirror, multilayer mirrors, or a polarization-selective rotator. Backward waves at selected frequencies have constructive interference, while forward waves have destructive interference. The backward constructive interference occurs in a narrow range of wave- length around the Bragg condition λB = 2neffΛ, where neff is the effective refractive index of the core. The strong and weak grating limits are defined as ΔnL ≫ λB and ΔnL ≪ λ B, respectively, where Δn = n − n0 is an index modulation and L is the FBG length. For the strong and weak FBGs, the reflection bandwidth is proportional to Δn and 1/L, respectively. The FBG bandwidth is typi- cally below one nanometer. In the polarization rotator, a mode with a given polarization is coupled to another mode with a different polarization. The conservation laws of energy and momentum can be employed to obtain the governing equations of FBG interferometer. The period of FBG depends 50 Optical Fiber Sensors: Advanced Techniques and Applications Light source Detector OFC p oo nac fiber l CCW CW Sag FIGURE 3.7 A schematic of Sagnac interferometer. on its application and can vary systematically or randomly along the optical fiber core. In a chirped FBG, the period varies monotonically along the optical fiber and has many applications in sensors and optical fiber networks. 3.4.3 Sagnac Interferometer A schematic of Sagnac OFI is presented in Figure 3.7. A single-mode stabilized semiconductor laser or erbium-doped optical fiber laser is employed as a light source for the interferometer. The laser beam is well collimated with uniform phase and splits into two parts with equal intensity by a 3 dB OFC. The two parts travel around a single-mode optical fiber coil (Sagnac coil) in opposite direc- tions. The output of the Sagnac coil is guided toward a single detector. The CW and CCW modes are in phase in a nonrotating fiber Sagnac interferometer, while in a rotating one due to the rotation velocity, the optical path of one of the modes is shorter than the other one. The interference spectrum depends on the angular frequency of the interferometer. Theoretical analysis is based on the Doppler shifts of the CW and CCW modes. It is assumed that the rotational axis is oriented along the optical fiber’s coil axis. The phase difference between the CW and CCW mode Δϕ versus the free- space laser wavelength λ, the coil area A, the number of coil turns N, and the coil angular frequency Ω is Δϕ = 8πNAΩ/(λc) [41,42]. The sensitivity S = Δϕ/Ω = 8πNA / ( λc ) increases by increasing the ( ) coil radius, total fiber length, and laser frequency. It is obvious that the total fiber length is restricted by the optical fiber attenuation and the coil radius is limited by the packaging size. Since the Sagnac interferometer’s output is independent of source noise, a superfluorescent ­optical source can be employed instead of a narrow linewidth laser source. Minimum polarization fading is another advantage of the Sagnac interferometer. The Sagnac interferometer fiber is insen- sitive to low frequencies. The Sagnac interferometer has been used for rotation sensing primarily. An optical gyroscope based on the Sagnac interferometer is commercially available. Fiber Sagnac interferometers are also employed for detecting current, acoustic wave, strain, and temperature [43–47]. Even a single-­ photon Sagnac interferometer with a net visibility of up to 99.2 ± 0.04% for gyro and quantum application was reported. 3.4.4 Fiber Ring Resonator Interferometer As shown in Figure 3.8a, a ring resonator is a fiber ring which is coupled to the input–output fibers through an optical fiber directional coupler. At frequencies where the incoming light and coun- tered fields are in-phase at the input of the ring resonator, constructive interference causes field Interferometric Fiber-Optic Sensors 51 Light OFC1 source Light Detector1 OFC source Detector OFC2 Detector2 (a) (b) FIGURE 3.8 A schematic configuration of (a) fiber ring resonator and (b) double-coupler fiber ring resonator. enhancement proportional to the fiber ring quality factor. In lossless optical fibers, the absolute value of transmission coefficient of the ring resonator is unity at all frequencies, while its phase abruptly changes around the resonance frequency. In other words, a lossless ring resonator is an all-pass filter, while for a lossy optical fiber at the resonance frequency, the field in the ring resona- tor increases and the output decreases. Commonly, in ring resonators, the bandwidth is inversely proportional to the fiber ring quality factor. One of the advantages of fiber ring resonator interferometer in comparison with other feedback fiber devices—such as FP interferometer and FBG—is that a multiple input/output design can eas- ily be achieved. For instance, Figure 3.8b shows the structure of a double-coupler ring resonator, in which the input is coupled through coupler 1 and the output can come from coupler 1 or 2. The output of OFC2 has its maximum value at the resonance frequency. Therefore, the lossless double- coupler fiber ring resonator has the advantage of not depending on phase measurement compared to fiber ring resonator. The resonance frequency depends on the fiber length and its refractive index (nL = mλ; m ∈ ). The length of ring resonators can be from several meters to a few micrometers. Microrings are usually used in photonic integrated circuits. By exploiting nonlinearity (such as Kerr effect or saturable absorption) in the fiber ring resonator, various types of instabilities (such as bistability, monostability, periodic pulse generation, optical turbulence, and chaos) can occur in fiber ring resonator, given suitable conditions [49,50]. This wide range of dynamic behavior makes fiber ring resonators a multipurpose element (having applications in clock pulse generation, timing control, and all-optical memory) for utilization in optical fiber communication networks. 3.4.5 Mach–Zehnder Optical Fiber Interferometer The Mach–Zehnder optical fiber interferometer (MZOFI) is so flexible that it can be employed for many diverse applications. An isolated laser diode is employed as the light source of long coherence length. In the two-legged MZOFI, the light is split into two similar parts by a symmetric OFBS and coupled to the two legs of MZOFI. Typically, the difference of the optical path lengths can be detected by a homodyne demodulator. The N-path MZOFI can be easily constructed using N × N couplers and single-mode optical fibers. Figure 3.9a shows an N-path Mach–Zehnder interferometer. Each 2N-port coupler in the presence of parasite reflection and polarization is described by a 4N × 4N unitary matrix. For linear polarized (LP) fields and in the absence of polarization changing devices, the coupler can be char- acterized by an N × N matrix. The tritters that are commercially available are described by a 3 × 3 unitary matrix. For LP modes propagating in single-mode fibers, a scalar analysis is sufficient. As an example, a three-path Mach–Zehnder interferometer is described by the product of two cou- pler matrices  and an optical path matrix  = diag eiϕ1 , eiϕ2 , eiϕ3 via M = , where φi(i = 1,2,3) ( ) is the phase of the ith path. When only one of the input fields is nonzero Ein = ( Ein , 0, 0 ) , the output ( ) 52 Optical Fiber Sensors: Advanced Techniques and Applications PS1 Detector PS2 Light PS3 OFC OFC source PSN (a) (b) (c) (d) (e) (f) (g) FIGURE 3.9 A schematic of (a) N-path MZOFI and (b–g) various configurations of in-line MZOFI. fields are obtained by the Mach–Zehnder transformation matrix (Eout = MEin). The output intensities In (n = 1,2,3) versus the optical path phase differences φij = φi − φj (i,j = 1,2,3) and input intensity I0 are given by I I n =  0 ⎡⎣3 + 2 cos ( ϕ12 + θn ) + 2 cos ( ϕ23 + θn ) + 2 cos ( ϕ31 + θn ) ⎤⎦ n = 1, 2, 3 (3.27) 9 where (θ1, θ2, θ3) = (0, −2π/3, 2π/3). In the presence of loss or gain in optical fibers, the optical path matrix is  = diag(a1, a2eiϕ12, a3eiϕ13 ) , where an(n = 1, 2, 3) is the transmission coefficient of the nth optical fiber branch. In this case, the output intensities are obtained by I0 2 In = ⎡ a1 + a22 + a32 + 2a1a2 cos ( ϕ12 + θn ) + 2a1a3 cos ( ϕ13 + θn ) + 2a2 a3 cos ( ϕ23 + θn ) ⎤ (3.28) 9 ⎣ ⎦ In the interference pattern of an N-path OFI, N−2 side lobes are observed between the main peaks. This is similar to the interference pattern of an N-slit illuminated by a monochromatic plane wave. Since the slopes and main peaks are steeper in an N-path interferometer, its sensitivity is higher than a conventional MZOFI. In addition, due to the phase differences φij being sensitive to the environ- mental parameters such as temperature and strain, MZOFI is a suitable choice for environmental parameter measurement. Due to the higher sensitivity of the cladding modes to the changes in the surrounding environ- ment, conventional MZOFI are replaced by in-line waveguide MZIs in which the core and cladding modes are employed as the two arms of the interferometer [52,53]. As shown in Figure 3.9b–g, sev- eral configurations for the in-line MZOFI have been proposed and investigated, both theoretically and experimentally. In-line waveguide MZIs use different methods to couple core modes to the cladding and then recouple them to the core. The recoupled cladding mode creates an interference with the uncoupled core mode, which makes this structure very compact and efficient. Interferometric Fiber-Optic Sensors 53 As shown in Figure 3.9b, a part of the mode guided in the single-mode fiber is coupled to the cladding modes of the fiber via an LPG and then recoupled to the core mode by another LPG. Due to modal dispersion, the core and cladding modes have different optical path lengths. The recombination of core and cladding mode in the core produces an interference pattern [53–59]. The LPG-Mach–Zehnder interferometer can be employed for multiparameter measurement. As shown in Figure 3.9c, the core–cladding mode splitter can be made by a very small lateral offset of the fiber cores. Due to the fiber offset, a part of the core mode is coupled to the cladding modes and then recoupled to the core mode by a second core–cladding mode splitter. The coupling coefficient of offset fibers is nearly wavelength independent; hence, the offset method can be employed at any wavelength and is cost-effective compared to the in-line MZI composed of a pair of LPGs. The in- line MZOFI can be simply made by a commercial fusion splicing. The cladding mode and insertion loss can be controlled by the amount of lateral offset. This lateral offset can be adjusted so that only one cladding mode is dominant. As shown in Figure 3.9d, another proposed method for in-line MZOFI manufacturing is collaps- ing air holes of a PCF, which does not need any alignment or cleaving processes. The PCF mode is expanded at the collapsed region and part of its energy is coupled to the cladding modes. In this structure, the coupling to several cladding modes is observed and controlling the number of clad- ding modes is not so simple. The insertion loss of this structure is high in comparison to the offset method, although by combining the LPG and collapsing methods, this insertion loss can be reduced. Fibers with different core sizes can be used for beam splitting [62,63]. One method is splicing a short piece of multimode fiber between two single-mode fibers as shown in Figure 3.9e. The light exiting the single-mode fiber is spread at the multimode region and then coupled into the core and cladding of the next single-mode fiber. A small core fiber can be inserted between two conventional single-mode fibers to make an in-line MZI. As shown in Figure 3.9f at small core fiber region, part of the light is guided as a cladding mode. An in-line MZOFI can be obtained by tapering a single-mode fiber at two points along the fiber, as shown in Figure 3.9g [64,65]. At the tapering points, the core mode diameter increases and part of it couples to cladding modes. This structure is very simple but the tapering regions are mechanically weak. Many other configurations for in-line MZOFI such as using double cladding fiber , microcavities , and a twin-core fiber have been investigated. 3.4.6 Michelson Optical Fiber Interferometer As shown in Figure 3.10a, the fabrication method and operation principle of conventional Michelson OFI (MOFI) is very similar to the MZOFI. The main difference is the mirrors at the end of the interferometer legs, which cause the MOFI to become a folded MZOFI. In a conventional MOFI, the high coherent light is split into two optical paths by a 2 × 2 OFC. The reflected light by mirror M1 and M2 are recombined by the OFC to produce interference patterns at the detector. The compact in-line configuration of MOFI is also possible, which is depicted in Figure 3.10b. Part of the core mode is coupled to the cladding modes by core–cladding mode splitter. Both the core and cladding modes are reflected by a common reflector at the end of the fiber [69–72]. The LPG can be used as a core–cladding BS in the in-line MI structures. For some applications, to prevent the environmental effects on LPG behavior, metal-coated LPGs are employed. To reduce the temperature effect on the measurements, fused silica PCF can be employed. As shown in Figure 3.10c, the conventional MOFI can be generalized to an N-path MOFI. The generalized BS is characterized by an N × N matrix (U) in a scalar model. The optical path matrix and the mirror reflection matrix are  = diag(e − iϕ1 ,…, e − iϕN ) and M = diag(−1, −1, …, −1), respec- tively. The output fields are related to the input fields by the scattering matrix (S = UMU ) via Eout = SEin, where Ein = E (f1) , E (f2 ) ,, E (f N ) and Eout = Eb(1) , Eb( 2 ) ,, Eb( N ) are the input and output ( ) ( ) 54 Optical Fiber Sensors: Advanced Techniques and Applications PS1 M1 Light source OFC PS2 M2 Detector (a) Mirror (b) PS1 M1 PS2 M2 PS3 Light OFC M3 source PSN MN (c) FIGURE 3.10 A schematic configuration of (a) basic MOFI, (b) compact in-line MOFI, and (c) N-path MOFI. fields and E (fi ) and Eb(i ) are the forward and backward fields in the ith optical fiber. It is assumed that only one of the input fields is nonzero. The input field vector is Ein = (ε1, 0, 0,…, 0) and the output intensities at the jth port Ij are as follows: 2 N −2 iϕm I j = ε1 ∑U m =1 U m1e jm (3.29) The sensitivity of the multipath MOFI is also greater than the conventional two-path ones. 3.4.7 Modal Optical Fiber Interferometer Different modes of multimode fibers have different velocities and the modal interferometers are established on the basis of this effect (dispersion). Typically, LP01 and LP11 modes or HE11 and HE21 modes of step index optical fibers can be employed to design the modal interferometers. Moreover, the two eigen polarizations of PMF can be employed for modal interferometry. The unique properties of holey fibers can be employed to design modal OFI. The PCF interferometers have the advantage of detecting, sensing, or spectroscopic analyzing of gas and liquids. The holey and hollow fibers have their own advantages. In holey fibers that are also called index-guiding fibers or solid-core PCFs, the desired gas or liquid interacts with cladding evanescent fields that have a few percentage of the total light power, while in HC-PCF, the gas or liquid interacts with the central part of the fundamental mode, which is more than 90% of the total light power. The bandwidth of Interferometric Fiber-Optic Sensors 55 silicon-core single-mode PCF is more than one thousand nanometers, which is greater than those of an air-core PCF fiber. A nanolayer of rare metal coating on the surface of the core and voids causes plasmon–light interaction in PCF and extremely enhances the interferometer’s sensitivity. Depending on the modal fiber interferometer, the birefringent PCF or PANDA fiber can be employed. 3.4.8 Moiré Optical Fiber Interferometer Overlapping of two or more gratings at different angles (θ) creates fringe patterns, which are the basis of Moiré interferometry. The desired fringe pattern can be achieved by choosing a suitable arrangement of optical fibers. N polarization-maintained optical fibers are employed for generation of interference grid pattern. The polarization angle of the jth fiber relative to the x-axis is denoted by θj (j = 1, 2,…, N) and its center coordinate in the z = 0 plane is (aj,bj). The field in the z = D plane at the point (x, y) is given by N ⎡k  ⎤ − i ⎢ ( xa j + b j y ) + ϕ j ⎥ E ( x, y ) = ∑ E j (a j , b j )e ⎣ D ⎦ + c.c. (3.30) j =1 where φj is the phase of the field of the jth fiber at z = 0 plane. The field intensity at the (x,y) point in the observation plane versus light intensity corresponding to the ith fiber (Ii (i = 1,2,…,N)) is N ⎧k ⎫ I= ∑I + ∑ i I i I j cos(θi − θ j ) cos ⎨ ( ai − a j ) x + ( bi − b j ) y − ϕ ij ⎬ (3.31) i =1 i≠ j ⎩ D ⎭ where φij is the phase difference between the ith and jth optical fiber k is the incident light wave number The desired fringe pattern can be obtained by choosing suitable fiber coordinates and polarizations. As an example, consider a system of three fibers centered at P(0,0), P(2a,0), and P(0,2a), where a is the radius of the PMF. The fibers’ arrangement and interference patterns are shown in Figure 3.11a. The horizontal and vertical patterns correspond to the interference of fibers 1 and 3 and fibers 1 and 2, respectively. The oblique lines in Figure 3.11a are due to the interference of fibers 2 and 3. By employing perpendicular polarizations for fiber 2 and 3, the interference between them cancels and oblique lines are eliminated. Setting an angle of 45 degrees between the polarization of fiber 1 and the x-axis ensures the occurrence of interference between both fibers 2 and 3 with fiber 1. To obtain a desired interferometric pattern, an arbitrary configuration of PMF is chosen with vari- able positions ((ai, bj) and θj). In order for the generated intensity distribution to be the closest to the desired distribution, in the sense of the metric of the L2 (R2) space, the parameters ai, bj and θj must be chosen wisely. 3.4.9 White Light Optical Fiber Interferometer In coherent light interferometry, the coherent length of the narrowband sources such as lasers is much greater than the optical path length difference in the interferometers. The fringes have a periodic structure, so the interferometric measurement suffers from an integer multiple of 2π phase ambiguity. Hence, coherent interferometry does not produce absolute data unless extra complexity is added to the interferometer itself. Using wideband light sources, the phase ambiguity is elimi- nated. Such an interferometry is called low coherency or white light interferometry (WLI). 56 Optical Fiber Sensors: Advanced Techniques and Applications Fiber 2 Fiber 2 Fiber 1 Fiber 3 Fiber 1 Fiber 3 (a) (b) FIGURE 3.11 (a-b) Two specific arrangements of (top) fiber-optic Moiré interference system and (bottom) their corresponding interference patterns. In WLI, corresponding to each wavelength, a separate fringe system is produced. The fringes of different wavelengths will no longer coincide as moving away from the center of the pattern. The electric field at any point of observation is the sum of electric fields of these individual patterns. The overall pattern is a sequence of colors whose saturation decreases rapidly. The WLI is adjusted such that the OPD is zero at the center of the field of view, so the electric field of different wave- lengths exhibits a maximum at the center point. So by measuring the fringe peak or envelop of the interferogram, the phase difference can be obtained without any ambiguity. The light sources such as tungsten lamps, fluorescent lamps, super-luminescent diodes (SLDs), light-emitting diode (LEDs), laser diodes near threshold, and optically pumped erbium-doped fibers can be used in WLIs. The spectral width of SLD and LED is between 20 and 100 nm and coherence length is less than 20 μm at the operating 1.3 μm wavelength. Generally, WLI operates on the basis of balancing the two arms of the interferometer by com- pensating for the OPD in the reference arm. The length of the reference arm can be controlled by different methods such as moving mirrors or piezoelectric (PZT) devices. The intensity of the interference fringe drops from a maximum to a minimum value by increas- ing the OPD between the two paths of WLI. Measuring the position of the central fringe is of utmost importance in WLI. The distance between the central fringe and its adjacent side fringe is so small that in the presence of noise there are some ambiguities in determining the central fringe position. These ambiguities can be removed by employing a combinational source of two or three multimode laser diodes with different wavelengths [81,82]. WLFIs are designed on different topologies of single- or multimode fiber interferometers. Single- and multimode fibers have their own advantages and disadvantages. White light single-mode fiber interferometer provides stable and large signal-to-noise ratio, whereas interferometers based on the multimode fibers employ cheaper optical components [81,82]. There are several WLFIs correspond- ing to each of the standard OFIs or their combinations [83,84]. Figure 3.12a shows a WLFI based on the MOFI working in the spatial domain. LED light splits by a lossless 2 × 2 OFC and couples to the arms of the MOFI. The reflected beam recombines on the avalanche photodiode (APD) of the MOFI. The mirror M2 is adjusted to a maximum output corresponding to the position of the central fringe. Interferometric Fiber-Optic Sensors 57 M1 Iin Iin LED ω ω OFC APD Movable mirror M2 (c) (b) (a) FIGURE 3.12 (a) A schematic configuration of WLI Michelson interferometer, (b) the spectrum of a normal LED (up) and the corresponding fringe pattern of the WLFI (down), and (c) the spectrum of three peak LED (up) and the corresponding fringe pattern of the WLFI (down). Figure 3.12b shows the fringe pattern of the WLFI based on the MOFI, for OPDs less than coher- ent length of the source. The position of the highest amplitude corresponds to exactly zero OPD. In Figure 3.12c, the results of three peak LEDs are compared with those of a normal LED to see how the precision increases when the multiwavelength wideband light source is employed in comparison to the single-wavelength wideband interferometer. 3.4.10 Composite Optical Fiber Interferometer Some first-order or field interferometers were already discussed in Sections 3.4.1 through 3.4.9. In all of the mentioned interferometers, the OPD between two or among many paths can be measured on the basis of field interference. Evidently, combination of different topologies can produce a new topology with advantageous properties. For instance, Michelson–Sagnac inter- ferometer has new properties for applications in quantum optomechanics. The reentrant topologies have feedback loops that have considerable effect on the stability of the interferome- ter. For instance, the double-loop interferometer such as that shown in Figure 3.13 has phase and polarization stability. By employing the proper loop’s parameters, the fiber interferometer operates in the stable regime. The higher-order interferometers are not in the scope of this chapter. However, Hanbury HB-T interferometer that is a well-known configuration for second-order interferometry was introduced briefly in Section 3.2.2. As an example, the intensity interferometry can be used in astronomy to measure the angle between the light sources such as stellar. SOA PC Att. 1 4 Detector Light 2 5 OFC source 3 6 FDL PC FIGURE 3.13 A schematic configuration of double-loop OFI. FDL, PC, Att., and SOA stand for fiber delay line, polarization controller, attenuator, and signal optical amplifier, respectively. 58 Optical Fiber Sensors: Advanced Techniques and Applications 3.5 SIGNAL RECOVERING AND NOISE SOURCE IN OPTICAL FIBER INTERFEROMETRY Different configurations of interferometric fiber sensors can be classified into balanced and unbal- anced paths design. The balanced-paths interferometer can be made by making the reference and sensing arm similar to each other by employing some devices such as PZTs. The balanced scheme is very popular because it usually does not require high-coherence sources. The imbalanced-paths configuration is very sensitive to laser noise and requires a very long coherence length laser. In the unbalanced fiber-optic interferometric sensors, the phase difference is measured by signal recovery method. The accuracy of OPD measurement by different optical fiber interferometry is limited by the noise sources. In this section, the signal recovery methods and sources of noise in OFIs are briefly reviewed. 3.5.1 Signal Recovery Method As mentioned in Section 3.4, the OPDs are converted to the optical phase differences in different types of optical interferometers. The phase difference variation has sufficiently low-frequency components. The phase difference can be measured both in time and spatial domain. When the phase differences are converted to light intensity on the spatial observation plane, then the pho- todiode matrix or charge-coupled devices (CCDs) can be employed to measure the interferomet- ric fringe pattern. Generally, the outputs are proportional to the cosine of the phase difference (cos ϕ). In the absence of sin ϕ, there is a π radian ambiguity in the phase recovery. If the phase difference is greater than 2π radians in addition to the sine and cosine values, the history of phase variation must be tracked to obtain the precise phase difference and corresponding OPD. Methods based on the production of new frequencies are called heterodyne detection; other meth- ods are named homodyne. The three main recovering methods described in the following sec- tions are phase-generated carrier homodyne detection, fringe-rate measurement, and homodyne detection methods. 3.5.2 Phase-Generated Carrier Homodyne Detection The semiconductor laser source of the interferometer is driven by a biased sinusoidal current. The laser frequency and its output power are modulated by the laser current variation. Corresponding to given OPDs, the phase differences at the output of the interferometer changes by varying the laser output wavelength. Therefore, the current frequency is modulated to the photo diode output. The current frequency and its harmonics can be observed in the side bands of the demodu- lated outputs. Two copies of sidebands are chosen by band-pass filters. Choosing filters with proper characteristics guaranties copies with the same amplitude. The filter outputs are employed as the inputs of an electronic mixer. The two outputs of the mixer are proportional to the cosine and sine of the phase difference of interest. Sometimes, the method is also called pseudoheterodyne detec- tion (PHD). In synthetic heterodyne detection , instead of laser frequency modulation, the phase shift modulation is produced by wrapping one arm of the interferometer around a PZT device, which is driven by a sinusoidal voltage. 3.5.3 Fringe-R ate Method In order to measure phase shifts greater than 2π radians, the fringe-counting and fringe-rate demod- ulation methods are introduced [89,90]. These methods are based on the transition of interferomet- ric output across some central value. The fringe-counting methods are based on the digital counting of the detector output in a suitable period of time. The frequency is the ratio of the counting number Interferometric Fiber-Optic Sensors 59 to the counting time. Due to the Heisenberg’s inequality principle, the resolution of frequency is proportional to the inverse of the counting time. The phase difference can be obtained by integrat- ing the instantaneous frequency. The fringe-rate method is based on a frequency to voltage converter (FVC). The detector output is employed as an input of FVC. The FVC output is integrated to obtain the phase difference. The minimum detectable signal is of the order of π radians and can be improved by axillary circuits. 3.5.4 Homodyne Method As mentioned in previous sections, the range of phase difference measurement by the fringe-­ counting and fringe-rate methods have a lower limit of π radians, while that of synthetic hetero- dyne method has an upper limit of π radians. The combination of both methods has no limitation on the phase difference measurement. In principle, the phase difference of different paths is pre- cisely m­ easurable, since the outputs of the OFIs are related to the input fields through a unitary matrix and unitary matrices are invertible. A number of homodyne techniques are used to bridge the gap between the measurement range of synthetic heterodyne detection method and that of fringe-­ counting and fringe-rate methods. OFIs usually have two outputs. In the homodyne method, an external 2 × 2 OFC is employed to create the interference patterns from the OFI’s outputs. Due to the unitarity of the scattering matrix, it is easy to show that the two outputs of OFC are 180° out of phase from one another. Hence, when all energy is presented in one of the OFC’s outputs, the other one is completely dark and vice versa. Therefore, no orthogonal components can be found in the OFC’s output. The orthogonal compo- nent can be produced by the heterodyne method. In order for the orthogonal component to directly appear in the outputs of the external coupler, a generalized OFC can be employed. For example, a 3 × 3 coupler is used as the external coupler of interferometer to create outputs with orthogonal components without employing the heterodyne detection technique. 3.5.5 Noise Sources in OFIs The precision and the minimum detectable phase differences of OFIs are determined by their output signal-to-noise ratio. The sources of noise can be originating from the light source, optical fibers, electronic circuits, and the surrounding environment. In practice, the output noise also depends on the OFI topology. In other words, any random process taking place in each part of the OFI acts as a noise source in it. Laser is the source of light in narrowband OFIs, while LED or any other wideband light gen- erator is used as a light source for WLOFIs. Laser sources operate on stimulated emission, while wideband light generators are based on spontaneous emission of atoms and molecules. The spon- taneous and stimulated emission of atoms and molecules are quantum effects, which are stochastic processes and can produce phase and amplitude noise in the light source output [92–94]. On the other hand, the interaction between laser cavity and its surrounding heat bath modes through the mirror coupling is another noise source for the laser output. The cavity as a narrowband filter reduces the laser noise significantly [96,97]. The presence of phase and amplitude noise causes the bandwidth of laser light to increase. By employing proper cavity in a single-mode laser, the band- width can be reduced to several kilohertz corresponding to several tens of kilometers for coherence length. Cross-saturation and mode competition effects are new noise sources in multimode lasers that can make multimode laser more suitable for WLFI than the narrowband OFI. Rayleigh scattering, Mie scattering, core–cladding interface scattering, absorption, amplifica- tion, Brillouin scattering, and Raman scattering are the main sources of noise in OFI arms and connectors. The first five cases are linear and the two latter ones are nonlinear noise sources. Some fraction of light that is deviated by scattering is trapped in the guided region and can propagate in both forward and backward directions along the optical fiber axis. This fraction of scattered light 60 Optical Fiber Sensors: Advanced Techniques and Applications contributes to phase and amplitude noise simultaneously. Other parts that are scattered out of the optical fiber affect the amplitude noise only. The nonlinear effects such as Brillouin and Raman scattering have two different components: Stokes and anti-Stokes frequencies. The Brillouin and Raman shifts are due to the light interaction with acoustic (± 25 GHz) and optical phonons (13 THz), respectively. Beating between Stokes, anti-Stokes with laser light cannot be observed at the output of any realistic detector and will be filtered intrinsically by the detector as a low pass filter. The Brillouin and Raman nonlinear losses are considered as a source of amplitude noise. The Raman, Brillouin, and Rayleigh scattering are symmetrically distributed with respect to forward and backward propagation directions, while those of Mie and core–cladding interference scatter- ing are mainly in the forward propagation direction. In multimode fiber interferometers, the mode coupling is another source of noise. APD, PIN diode, CCD, and photomultiplier (PhM) are used as an electronic detector in OFIs. Thermal noise, dark current noise, shot noise, background noise, and flicker noise are common noises in all optical detectors. The electron–hole generation and recombination are stochastic pro- cesses and are noise sources in semiconductor detectors. The avalanche effect is also a random process and is a noise source in APDs. The same effect can be found on the anodes of PhMs and create noise in the detector. The Johnson noise, shot noise, burst noise, and flicker noise of different electronic elements are the final intrinsic noise of OFIs. The optical fiber parameters can be affected by the environmental physical variations such as mechanical vibration, acoustic agitation, pressure, tension, and thermal variations. These effects are channels for transferring environmental fluctuations to the OFI output as noise. In conclusion, the output signal is affected by all noise sources. The output signal can be denoised by signal processing methods. Depending on the signal’s behavior, an appropriate denoising method must be used. For instance, the windowed Fourier method can be employed to denoise the music-like signals of a fiber intruder detector that operates based on the birefrin- gent fiber interferometer. Wavelet analysis methods such as Fourier regularized deconvolu- tion (FoRD) and Fourier-wavelet regularized deconvolution (ForWaRD) are employed to denoise the transient signals. 3.6 INTERFEROMETRIC OPTICAL FIBER SENSORS (IOFSs) The optical path length of each arm of the OFI—which is the product of fiber refractive index and geometrical length of the arm—can be perturbed by one or more environmental physical parame- ters. The optical path length variation can be measured precisely by the optical fiber interferometry. This is the basic concept of IOFS operation. By measuring the phase difference, one can sense the changes in the environmental physical parameter. 3.6.1 Principle of Operation of IOFSs The IOFSs can be classified into resonance and nonresonance types. The resonance types such as optical fiber FPIs, ring resonators, and FBGs are operating on the basis of constructive and destruc- tive interference. In the resonance-based IOFSs, the resonance frequency changes as a function of the environmental physical parameters. The resonance frequency or bandwidth measurement can be employed to sense the environmental parameter changes. The output amplitude or phase change at a given frequency can provide some information about IOFSs’ environment. The nonresonance IOFSs are operating on the basis of comparing the OPD of the different arms of the OFI. One or several arms are kept isolated from external variations and only the sensing arm is exposed to the environmental variations. The OPD can be easily detected by analyzing the variation in the interference signal. The phase of resonators is highly sensitive to the optical length variation in the vicinity of resonance frequency. Hence, employing resonators in the sensing arm of the nonresonance OFI improves the sensitivity of the sensor versus the environmental variations. Interferometric Fiber-Optic Sensors 61 In some applications such as chemical and biological sensors, cooperation of plasmons in the sensing arm causes an enhancement in the sensitivity of the IOFSs. 3.6.2 Principle of Plasmon When an incoming electric field is coupled to the collective oscillations of free electrons on a metal surface (surface plasmon resonances), the surface electromagnetic waves that are called sur- face plasmon polaritons (SPPs) are excited and propagate along the metal–dielectric interface. SPP modes can be excited at the flat and curved metal–dielectric interfaces such as metal nanostruc- tures. The SPPs are able to concentrate electromagnetic waves beyond the diffraction limit and enhance the field strength by several orders of magnitude [99,100]. The simplest geometry for sustaining SPPs is the flat interface between a dielectric and a con- ductor. The relative permittivity (or dielectric constant) of dielectric medium is denoted by εd. The dielectric constant of a conductor depends strongly on the incident light’s frequency and is called the dielectric function εm(ω). Since TE waves (S-polarized) are purely transverse waves, they cannot be coupled to the longitudinal electron oscillations in metal and only TM (P-polarized) waves can excite the SPP waves at the metal–dielectric interface. By applying the boundary condi- tions at the interface, the required condition for exciting SPP modes can be obtained directly. For εd > 0, the ­confinement of electromagnetic wave at the conductor–dielectric interface requires that Re{εm(ω)} < 0. This condition is satisfied at frequencies below the bulk plasmon frequency (ωp). In lossless metals with Re{εm(ω)} < 0 because of the pure imaginary wave vector, there are no propagating waves k x2 + kz2 = ε m ( ω) k02 < 0. Typically, εm is much greater than one ε m  1 , which ( ) ( ) −1 ( ) means the penetration depth in metals kz ≅ λ 0 / 2π ε m is rather small and can be on the order of nanometer scale. By introducing a TM-polarized electric field as E ( z > 0 ) = E x0 , 0, E zd ⋅ ei( ksp x − ωt ) ( ) ( 2 sp 2 d 0 ) exp − z k − ε k and E ( z < 0 ) = E , 0, E ⋅ e ( 0 x ) i ( ksp x − ωt ) exp − z k − ε k into Helmholtz wave m z ( 2 sp 2 m 0 ) equation and employing the boundary conditions at the metal–dielectric interface, one can derive the SPP dispersion relation as follows: ω εd εm ( ω) ksp ( ω) = (3.32) c ε d + ε m ( ω) The necessary conditions for an SPP excitation are Re{εdεm(ω)} < 0 and ε d + Re ε m ( ω) < 0. In the { } long-wavelength range of the visible spectrum and in the infrared region, these conditions are satis- fied for most metals and dielectrics. The other key point is the SPP wavelength λsp = 2π /Re(ksp), which is always smaller than the incident light wavelength in the dielectric. The damping of SPPs corresponding to ohmic loss of electron oscillations is a very important factor that restricts both the lower limit of SPPs’ wavelength and its propagation length [99,101]. The amplitude of normal and tangential electric field compo- nents in the dielectri