1.2 Types of optical fibers

The commonly known optical fibers are step index and graded index (GR-IN) optical fibers. The former means that the core's refractive index is homogeneous while it suffers an abrupt change at the boundary with the clad. For a GR-IN optical fiber, the core does not have a constant value of refractive index but it rather has a radial distribution of refractive index. These two types of optical fibers can be classified into either single-mode or multi-mode optical fibers. Single-mode optical fiber only sustains one mode of propagation while the multi-mode optical fiber can sustain up to hundreds of propagation modes [1, 3]. The number of the propagation modes is related to the numerical aperture of the fiber, which, in turn, depends on the refractive indices of both core and clad.

### 1.3 Characterization of optical fibers

Accurate characterization of optical fibers is required in order to know about their functions and performances. There are many methods of optical fibers characterization such as optical microscopy, electron microscopy, X-ray spectrometry, infrared spectroscopy, light diffraction, light scattering, optical interferometry, and digital holography [1, 3, 12–29]. Optical interferometry is an effective accurate tool for studying and characterizing optical fibers. It depends on the determination of the phase difference between a ray of light transmitting the fiber's cross-section and a reference ray reaching the interference plane directly without crossing the fiber. This phase shift can be transformed into a refractive index map representing the radial distribution of the refractive index across the fiber or, in other words, the refractive index profile (RIP). Interferometry can detect tiny changes in refractive index if an external effect is applied on the fiber. The change of refractive index can be in situ detected if the interferometer is developed to achieve this task. Interference patterns can be digitally processed and analyzed in order to increase the accuracy of the obtained results [1, 17, 22, 26, 27, 30–35].

Interference techniques can be classified into either two-beam interferometers such as Michelson, Mach-Zehnder, Pluta polarizing microscope, Lioyd's mirror, etc., or multiple-beam interferometers such as Fabry-Pérot and Fizeau interferometers [1, 3, 25, 36–39]. A two-beam interferometer produces a pattern of alternate bright and dark fringes of equal thicknesses when two beams, usually, of equal intensities Io suffering a relative phase difference δ are superposed. The resultant intensity distribution I of the interference pattern is given as:

$$I = I\_o \cos^2\left(\frac{\delta}{2}\right) \tag{1}$$

Multiple-beam interference takes place when light rays fall on two parallel optical plates enclosing a small distance between each other while their inner surfaces are highly reflecting and partially transparent. The intensities of both reflected, I (r), and transmitted, I (t), light distributions that are redistributed due to the multiple-beam interference are given as [40]

Optical Fibers Profiling Using Interferometric and Digital Holographic Methods DOI: http://dx.doi.org/10.5772/intechopen.91265

$$I^{(r)} = \frac{(2 - 2\cos\delta)R}{1 + R^2 - 2R\cos\delta} I^{(i)} \tag{2}$$

$$I^{(t)} = \frac{T^2}{1 + R^2 - 2R\cos\delta} I^{(i)} \tag{3}$$

where, I (i) is the intensity of the incident light,T and R are the products of the transmission and reflection coefficients of the two surfaces, respectively, while δ is the phase difference between any two consecutive interfered rays.

On the other hand, holography was firstly presented by Gabor in 1947 as a lensless process for image formation by reconstruction of wave-fronts [41–43]. It offers 3D characterization such as the depth of field from recording and reconstructing the whole optical wave field, intensity, and phase [41, 42]. Holographic interferometry is a non-destructive, contactless tool that can be used for measuring shapes, deformations and refractive index distributions [44, 45]. The modern digital holography was introduced in 1994 [46–48]. Moreover, the phase shifting interferometric (PSI) technique was introduced by Hariharan et al. as an accurate method for measuring interference fringes in the real time [49]. Recently, digital holographic phase shifting interferometry (DHPSI) was used to investigate some optical parameters of fibrous materials [17, 18, 21, 26–29].

#### 1.4 Digital holographic phase shifting interferometry (DHPSI)

In DHPSI, frequently a set of four [20] or five [23, 33] phase shifted holograms with known mutual phase shifts starting with 00 and having 900 separations have to be recorded [21]. These recorded holograms can be represented by:

$$I\_n = a(\zeta, \eta) + b(\zeta, \eta) \cos \left(\wp(\zeta, \eta) + \wp\_{\text{Ru}}\right), n = 1, 2, 3, \dots \tag{4}$$

where a(ζ, η) and b(ζ, η) are the additive and the multiplicative distortions and φRn is the phase shift of the reference wave. In case of four and five phase shifted holograms, the complex wavefield [37] in the hologram plane can be calculated using Eqs. (5) and (6), respectively.

$$h(\zeta,\eta) = [I\_1(\zeta,\eta) - I\_3(\zeta,\eta)] + i[I\_4(\zeta,\eta) - I\_2(\zeta,\eta)]\tag{5}$$

or,

$$h(\zeta,\eta) = \left[4I\_1(\zeta,\eta) - I\_2(\zeta,\eta) - 6I\_3(\zeta,\eta) - I\_4(\zeta,\eta) + 4I\_5(\zeta,\eta)\right] + i\left[7(I\_4(\zeta,\eta) - I\_2(\zeta,\eta))\right] \tag{6}$$

In digital holography, the recorded wavefield is reconstructed, based on Fresnel diffraction integral, by multiplying the stored hologram by the complex conjugate of the reference wave r\*(ζ, η) to calculate the diffraction field b'(x', y') in the image plane, see Figure 1. This can be calculated using the finite discrete form of the Fresnel approximation to the diffraction integral as:

$$b'(n\Delta\mathbf{x}', m\Delta\mathbf{y}') = \mathbf{A} \sum\_{j=0}^{N-1} \sum\_{l=0}^{M-1} h(j\Delta\zeta, l\Delta\eta) r^\* \left(j\Delta\zeta, l\Delta\eta\right)$$

$$\times \exp\left\{\frac{i\pi}{d'\lambda} \left(j^2 \Delta\zeta^2 + l^2 \Delta\eta^2\right)\right\} \exp\left\{2i\pi \left(\frac{jn}{N} + \frac{lm}{M}\right)\right\} \tag{7}$$

Figure 1. Geometry of digital holographic axes and the planes systems.

The parameters used in this formula depend on the used CCD array of N � M pixels and the pixel pitches Δζ and Δη. The distance between the hologram and the image plane is denoted by d'. The pixel spacings in the reconstructed field of image are:

$$
\Delta \mathbf{x}' = \frac{d'\lambda}{N\Delta \zeta} \text{ and } \Delta \mathbf{y}' = \frac{d'\lambda}{M\Delta \eta} \tag{8}
$$

The convolution of h(ζ, η)r\*(ζ, η) can be used as alternative of Fresnel approximation [37]. The resulting pixel spacing for this convolution approach is

$$
\Delta \mathfrak{x}' = \Delta \zeta \,\, \mathfrak{a} \\
\text{and } \Delta \mathfrak{y}' = \Delta \eta \,\, \tag{9}
$$

In addition, the phase shifted holograms are used to overcome the problems of the d.c. term and twin image, in which the calculated complex wavefield is used instead of a real hologram in the convolution approach.

The intensity and phase distributions in the reconstruction plane are given by

$$I(\mathbf{x}', \mathbf{y}') = \left| b'(\mathbf{x}', \mathbf{y}') \right|^2 \tag{10}$$

$$\rho(\mathbf{x}',\mathbf{y}') = \arctan\left\{\frac{\operatorname{Im}|b'(\mathbf{x}',\mathbf{y}')|}{\operatorname{Re}\left|b'(\mathbf{x}',\mathbf{y}')|}\right|\right\}\tag{11}$$

So, the optical phase differences due to phase objects can be extracted.

Mach-Zehnder interference-like system is used as a digital holographic setup as shown in Figure 2 [20, 23, 29, 33]. The optical waveguide sample, such as optical fiber, is immersed in a liquid of refractive index nL near or matching the cladding refractive index nclad of the sample. The interference patterns are recorded using a charge-coupled device, that is, CCD camera.

In this chapter, we illustrate some featured work on interferometric characterization (sometimes, implying digital holographic interferometry) of different optical fibers done by our research group during the last three decades. In Section 2, interferometric characterization of conventional step-index and GR-IN optical fibers is presented. Section 3 illustrates characterization of the conventional optical fibers when they are suffering mechanical bending. In Section 4, interferometric characterization of a special type of optical fibers called polarization maintaining (PM) optical fibers is presented. In the last section, we elucidate thick optical

Optical Fibers Profiling Using Interferometric and Digital Holographic Methods DOI: http://dx.doi.org/10.5772/intechopen.91265

#### Figure 2.

Mach-Zehnder digital holographic interferometric set-up, S F: spatial filter, L: collimating lens, BS: beam splitter, M: mirror, and MO: microscopic objective.

fibers and their interferometric characterization with a special interferometric system, developed in our laboratory, called lens-fiber interferometry (LFI).
