2. Conventional optical fibers

#### 2.1 Step-index optical fiber

In 1994, Hamza et al. derived a mathematical expression to calculate the RIP of an optical fiber by considering the refraction of optical rays at the liquid-clad and clad-core interfaces, see Figure 3 [12]. It was the first time to consider the refraction of the transmitted rays to reconstruct the RIP of a fiber. The derived expressions for calculating the RIP in case of two-beam and multiple-beam interferences, based on Figure 3, are given by Eqs. (12) and (13), respectively.

$$\begin{split} \frac{Z(\mathbf{x})\lambda}{h} &= 2n\_{\varepsilon} \left\{ \sqrt{\left[R^{2} - \left(\frac{n\_{L}d}{n\_{\varepsilon}}\right)^{2}\right]} - \sqrt{\left[\left(R - e\right)^{2} - \left(\frac{n\_{L}d}{n\_{\varepsilon}}\right)^{2}\right]} \right\} \\ &+ 2n\_{\varepsilon} \sqrt{\left[\left(R - e\right)^{2} - \left(\frac{n\_{L}d}{n\_{\varepsilon}}\right)^{2}\right]} - n\_{L} \left[\sqrt{\left(R^{2} - d^{2}\right)} + \sqrt{\left(R^{2} - x^{2}\right)}\right] \mathbf{0} \leq d \prec R \end{split} \tag{12}$$

$$\begin{split} \frac{Z(\mathbf{x})\lambda}{2h} &= 2n\_{\varepsilon} \left\{ \sqrt{\left[R^{2} - \left(\frac{n\_{L}d}{n\_{\varepsilon}}\right)^{2}\right]} - \sqrt{\left[\left(R - e\right)^{2} - \left(\frac{n\_{L}d}{n\_{\varepsilon}}\right)^{2}\right]} \right\} \\ &+ 2n\_{\varepsilon} \sqrt{\left[\left(R - e\right)^{2} - \left(\frac{n\_{L}d}{n\_{\varepsilon}}\right)^{2}\right]} - n\_{L} \left[\sqrt{\left(R^{2} - d^{2}\right)} + \sqrt{\left(R^{2} - x^{2}\right)}\right] \mathbf{0} \leq d < R \end{split} \tag{13}$$

where, R is the fiber's radius and e is the skin's thickness. nL, ns, and nc are the refractive indices of the immersion liquid, skin, and core, respectively. λ is the wavelength of the used illuminating source. Ls and Lc are the geometrical path lengths inside the skin and the core, respectively. Z is the fringe shift due to the presence of the fiber while h is the interfringe spacing and d is the distance measured from the center of the fiber to the position of the incident ray.

In that work, they used Fizeau interferometer to determine the refractive index profile of FOS Ge-doped step-index multi-mode optical fiber with a core radius 19.5 μm. The fiber was immersed in a liquid of refractive index nL = 1.4665, which was a little bit greater than ns while the wavelength of the used illuminating source was λ = 546.1 nm. The Fizeau interferogram of this fiber is shown in Figure 4a. The obtained RIP was compared with the profile calculated for the same fiber when the refraction of light through the fiber was neglected as was usually done by other authors before this work. There was a significant difference between the two profiles, see Figure 4b. Therefore, the refraction through the fiber was recommended to be considered for calculating RIPs particularly when the refractive index of the immersion liquid is not close to the fiber's refractive index.

In 2008, another mathematical model was derived in order to determine RIPs of fibers having regular and/or irregular cross-sections [38]. This method was based

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

#### Figure 4.

(a) An interference pattern of Fizeau fringes, in transmission, for a FOS step-index optical fiber. (b) RIPs of this fiber in case of considering and neglecting the retraction of the crossing rays inside the fiber.

on immersing the investigated fiber in two liquids with different, but so closed, refractive indices. They applied this method on a single-mode optical fiber, having a small core of radius <5 μm while the fiber's radius was 60.6 μm, as shown by Fizeau interferograms in Figure 5 when the fiber was immersed in two liquids with refractive indices (a) 1.4589 and (b) 1.4574. The obtained RIP of this fiber is illustrated in Figure 5c showing that this fiber has nc = 1.4630 and ncl = 1.4596. This method was simple and accurate enough to detect such a small core of a step-index optical fiber.

### 2.2 Graded-index (GR-IN) optical fiber

A GR-IN optical fiber with a radial refractive index distribution was suggested to be divided into a finite number (M) of concentric layers where each layer has its own value of refractive index, see Figure 6a. The thickness (a) of each layer equals R/M, where R is the radius of the graded-index part. When the ray falls on the fiber at a distance dQ apart from the fiber's center, the ray refracts through Q layers. The nearest layer to the fiber's center has a refractive index nQ. The fiber's RIP can be calculated using Eq. (14) in case of two-beam interference and Eq. (15) in case of multiple-beam interference [13]. Another model was presented in order to get RIP of a GR-IN optical fiber by considering the real path of the optical ray due to the

Figure 5.

Fizeau interferograms, in transmission, for a single-mode optical fiber when it was immersed in two liquids of refractive indices (a) 1.4589, (b) 1.4574. (c) RIP of the single-mode optical fiber having the interferograms shown in (a) and (b).

(a) A schematic diagram shows the path of an optical ray crossing Q layers in the core region. (b) A schematic diagram shows the path of an optical ray crossing a GR-IN core optical fiber.

refraction in the core region as well as adding a correction for the ray passing through the immersion liquid [50], see Figure 6b. In this case, the fringe shift was obtained by assuming values for both the profile shape parameter (α) and the difference between refractive indices of core and clad (Δn). A prepared software was programmed to iterate and get the best values of α and Δn and comapre the calculated fringe shift with the experimentally obtained one.

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

$$\begin{split} \frac{\lambda \mathcal{Z}\_{Q}}{h} &= \sum\_{j=1}^{Q-1} 2n\_{j} \left[ \sqrt{(R-(j-1)a)^{2} - \left(\frac{d\_{Q}n\_{o}}{n\_{j}}\right)^{2}} - \sqrt{(R-ja)^{2} - \left(\frac{d\_{Q}n\_{o}}{n\_{j}}\right)^{2}} \right] \\ &+ 2n\_{Q} \sqrt{(R-(Q-1)a)^{2} - \left(\frac{d\_{Q}n\_{o}}{n\_{Q}}\right)^{2}} - n\_{o} \left[ \sqrt{R^{2} - d\_{Q}^{2}} + \sqrt{R^{2} - \mathbf{x}\_{Q}^{2}} \right] \quad \text{(14)} \\ \frac{\lambda Z\_{Q}}{2h} &= \sum\_{j=1}^{Q-1} 2n\_{j} \left[ \sqrt{(R-(j-1)a)^{2} - \left(\frac{d\_{Q}n\_{o}}{n\_{j}}\right)^{2}} - \sqrt{(R-ja)^{2} - \left(\frac{d\_{Q}n\_{o}}{n\_{j}}\right)^{2}} \right] \\ &+ 2n\_{Q} \sqrt{(R-(Q-1)a)^{2} - \left(\frac{d\_{Q}n\_{o}}{n\_{Q}}\right)^{2}} - n\_{o} \left[ \sqrt{R^{2} - d\_{Q}^{2}} + \sqrt{R^{2} - \mathbf{x}\_{Q}^{2}} \right] \quad \text{(15)} \end{split}$$

According to Figure 6b, the optical pathlengths of the ray crossing the core Ol Kð Þ and the ray passing only in the immersing liquid OlL are given by the following relations.

$$\mathcal{O}l(K) = 2 \int\_{K}^{R} \frac{n^2(r)r}{\sqrt{n^2(r)r^2 - n^2(K)K^2}} dr \tag{16}$$

$$\mathcal{O}l\_L = \mathfrak{D}n\_o \mathcal{R} \frac{\sin\left(\varepsilon\right)}{\sin\left(\chi\right)}\tag{17}$$

Figure 7.

(a) Pluta duplicated image of LDF GR-IN optical fiber and (b) Fizeau interferogram of the same sample. Reference [50] with permission.

#### Figure 8.

A comparison between RIPs of LDF GR-IN optical fiber using the model in Ref. [27] (dots) and model in Ref. [28] (solid curve) in case of (a) multiple-beam Fizeau interference and (b) two-beam Pluta interference. Ref. [50] with permission.

where, R is the core's radius, k is the minimum distance between the fiber's center and the bent ray, ε is the half of the angle determined by the two radii that are enclosing the bent ray inside the graded-index region, and γ is the half of the angle between the incident and the emerged rays. Figure 7 shows the interferograms of LDF GR-IN optical fiber when it was investigated by (a) Pluta and (b) Fizeau interferometers. Figure 8 shows the RIPs calculated by these last models for the LDF optical fiber. The last model, presented in 2001 [50], provided more accurate values of the RIP of a GR-IN optical fiber compared with its previous presented model in Ref. [13].

However, the former requires knowing the function describing the index profile while the aim is to find the parameters of this function.
