3. Bent conventional optical fibers

Optical fibers, which are isotropic materials, can suffer a birefringence under external mechanical bending effects [1, 22, 33, 51]. The induced birefringence can be used in sensing applications [52–54]. However, bending has an unfavorable effect on the optical fibers used in telecommunications where it, sometimes, causes a mode disturbance and consequently a signal attenuation [55, 56]. An approach to calculate the refractive index profile of a bent optical fiber was proposed where the fiber was divided into layers and slabs simultaneously [22]. The refraction of the optical rays at the liquid-clad and clad-core interfaces was considered. Unfortunately, this approach did not consider the change of refractive index inside each slab. Also, the expected change of refractive index due to the release of stresses near the fiber's free surface has not been considered. However, this approach succeeded to present good information about the variation of mode propagation due to bending.

#### 3.1 Step-index bent conventional optical fiber

In 2014, Ramadan et al. calculated the refractive index and the induced birefringence profiles of bent step-index optical fibers using digital holographic Mach-Zehnder interferometer [33]. In that work, they considered two different processes controlling the variations of the refractive index of the bent fiber: (1) the linear refractive index variation due to the applied stress along the bent radius and (2) the release of this stress on the fiber's surface. The first one is dominant when approaching the center of the fiber while the second one is dominant near the fiber's free surface and decays on moving toward the fiber's center. Figure 9 shows the difference between the paths of optical rays through the bent fiber in the compressed and expanded parts. The stress release was supposed to have a radial dependence on the fiber's radius, which enabled the construction of 2D RIP of the investigated bent homogeneous optical fiber. Based on the expected stress values due to the bending effect, a function describing the RIP was proposed and used to integrate the optical path of the ray traversing the fiber [50]. By adapting the appropriate parameters of this function, the optical phase differences were estimated and matched those phase differences that were experimentally obtained. By this assumption, a realistic induced stress profile due to bending was obtained [33]. DHPSI was used in that study where the recorded phase shifted holograms were combined and processed to extract the phase map of the fiber [18]. By considering both of the mentioned effects, the following function was chosen to describe the RIP of the bent optical fiber [33].

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

$$n(\mathbf{x}, r) = -\rho n\_{bf} \frac{\mathbf{x}}{R} \left[ \mathbf{1} - e^{-\left(\frac{r\_Q - r}{r\_\gamma}\right)} \right] + n\_{cl} \tag{18}$$

where ρ is the strain-optic coefficient, nbf is the refractive index of the bent-free fiber, R is the radius of bending, ro is the radius of the fiber, ncl is the clad's refractive index, rs is the proposed parameter to control the distance suffering stress release from the surface of the fiber, and x is the distance between the center of the fiber and the position of the incident ray.

The first term of Eq. (18) gives the bent-induced birefringence,

$$
\Delta n(\mathbf{x}, r) = -\rho n\_{bf} \frac{\mathcal{X}}{R} \left[ \mathbf{1} - e^{-\left(\frac{r\_Q - r}{\eta}\right)} \right] \tag{19}
$$

which is correlated to the generated stress S (r,x) inside the fiber

$$S(r, \mathbf{x}) = E\left(\frac{-\Delta n(\mathbf{x}, r)}{\rho n\_{bf}}\right) = E\frac{\mathbf{x}}{R} \left[\mathbf{1} - e^{-\left(\frac{r\_Q - r}{r\_l}\right)}\right] \tag{20}$$

Eq. (20) evaluates the distribution of stress over the fiber's cross-section for different bending radii where E is the Young's modulus of the bent fiber. The signs of Δn are opposite to the signs of tensile and compressive stresses. The tensile stress was chosen to be positive.

Since bending such a step-index optical fiber converts it into a weekly gradedindex fiber, Bouguer's formula [40] was used to correlate the radius, incidence angles, and refractive index of the bent fiber as follows:

$$Kn(\varkappa, K) = rm(\varkappa, r) \sin \theta\_r \tag{21}$$

where n(x,r) is the refractive index at radius r. By applying this formula at the incidence point, one obtains

$$Kn(\varkappa, K) = r\_o n\_L \frac{\varkappa}{r\_o} = \varkappa n\_L \tag{22}$$

This equation was numerically solved to get K satisfying the lower integration limit of the optical path difference for a certain value of x. Based on the model described in Ref. [50], the infinitesimal change in the geometrical distance along the path of the optical ray with respect to the radius variation was given as:

$$\left(\frac{\partial l}{\partial r}\right)\_{n(\mathbf{x},r)} = \frac{2n(\mathbf{x},r)r}{\sqrt{n^2(\mathbf{x},r)r^2 - n^2(\mathbf{x},k)k^2}}\tag{23}$$

By integration with respect to r, the total path length inside the fiber is:

$$l(\mathbf{x}) = 2 \int\_{K}^{r\_o} \frac{n(\mathbf{x}, r)r}{\sqrt{n^2(\mathbf{x}, r)r^2 - (n\_L \mathbf{x})^2}} dr \tag{24}$$

The optical path length difference between this ray, passed through the fiber, and the reference ray passed through the liquid is:

$$
overline{(\infty)} = 2 \int\_{K}^{r\_o} \frac{\left(n(\infty, r) - n\_L\right) \cdot n(\infty, r)r}{\sqrt{n^2(\infty, r)r^2 - (n\_L \infty)^2}} dr \tag{25}$$

The phase difference is given as:

$$\phi(\mathbf{x}) = \frac{2\pi}{\lambda} opld(\mathbf{x})\tag{26}$$

Figure 10 shows a set of five shifted holograms of a bent step-index optical fiber with a bending radius R = 8 mm when the incident light was vibrating parallel to the fiber's axis. They were recorded in order to apply the DHPSI technique and reconstruct the RIP of the bent fiber. The 2π shifted interferogram was analyzed and its reconstructed interference phase map, enhanced phase map, and interference phase distribution are shown in Figures 11a–c, respectively. The refractive index crosssection distribution of the bent optical fiber is shown in Figure 12 while the strainoptic coefficients in compression and expansion were 0.208 and 0.224, respectively.

### 3.2 Graded-index bent conventional optical fiber

In 2017, Ramadan et al. presented a theory to recover the RIP of a bent GR-IN optical fiber inside the core region using DHPSI [35]. They assumed the two different processes controlling the shape of the RIP: (1) the linear variation due to stresses in the direction of the bent radius and (2) the release of the stresses near the fiber's surface.

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

Figure 10. A set of five shifted interferograms of a bent step-index optical fiber.

#### Figure 11.

(a) The reconstructed interference phase map modulo 2π, (b) its enhanced phase map, and (c) the interference phase distribution.

Figure 12. The refractive index cross-section distribution of the bent optical fiber, R = 8.

The total optical path length of the optical ray crossing the bent GR-IN optical fiber is given by Eq. (27), see Figure 13. The calculated optical path length differences of the interfered rays can be transformed, afterward, into a phase difference map using Eq. (26).

$$\text{OPID}(d) = \text{OPID}\_{cl}(d) + \text{OPID}\_{c}(d) \tag{27}$$

with,

$$OPlD\_{cl}(d) = 2\int\_{r\_c}^{r\_d} \frac{(n\_{cl}(d,r) - n\_L)n\_d(d,r)r}{\sqrt{n\_{cl}(d,r)^2r^2 - n\_{cl}(d,k\_{cl})^2k\_{cl}^2}} dr \tag{28}$$

$$OPlD\_c(d) = 2\int\_{k\_c}^{r\_c} \frac{(n\_c(d, r) - n\_L)n\_c(d, r)r}{\sqrt{n\_c(d, r)^2 r^2 - n\_c(d, k\_c)^2 k\_c^2}} dr \tag{29}$$

Figure 14a shows a set of five phase shifted interferograms for the bent GR-IN optical fiber with bending radius R = 8 mm when the incident light was vibrating parallel to the fier's axis. The enhanced reconstructed phase modulo 2π and the interference phase distribution of the bent fiber are shown in Figure 14b. Due to

Figure 13. schematic diagram shows the ray tracing in case of traversing bent GR-IN fiber.

Figure 14.

(a) A set of five phase shifted interferograms of a bent GR-IN optical fiber. (b) The enhanced reconstructed phase modulo 2π and the interference phase distribution. Ref. [35] with permission.

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

#### Figure 15.

Refractive index cross-section distribution of the bent GR-IN optical fiber when the incident light vibrates (a) parallel and (b) perpendicular to the fiber's axis. (c) The birefringence cross-section distribution, R = 8 mm. Ref. [35] with permission.

the bending process, the GR-IN optical fiber exhibited a birefringence where the RIPs when the incident light vibrated parallel and perpendicular to the fiber's axis were different, see Figure 15.
