5. Homogeneous thick optical fibers

Optical fibers having diameters in the order of 100 μm, or less, are convenient to be investigated using interferometric methods when the samples are put in immersion liquids of refractive indices close to the refractive indices of the fibers as

described in the previous sections [12, 13]. Optical fibers of diameters bigger than 150 μm cannot be investigated by normal interferometry where the planes of fringes in both liquid and fiber cannot be focused simultaneously. In 2000, Ramadan presented a novel interferometric method to recover such a problem for homogeneous thick optical fibers, commonly used in short-distance data transmission, without using immersion liquids [16]. This type of interference was called lens-fiber interferometry (LFI) since the interference fringes were produced by a combination of an aberrated cylindrical lens and a thick optical fiber. The aberrated cylindrical lens was used to focus a parallel beam on this fiber, which was located in the focal plane of the cylindrical lens [60], see Figure 22.

Two-beam interference produced by the superposition of two optical rays emerging from the fiber was recorded and explained. Due to the aberration of the cylindrical lens, one of these two rays crossed the thick fiber before its center while the other ray crossed after the fiber's center. Therefore, for each point in the image plane, two rays having two different initial incidence angles on the thick fiber are superposed, see Figure 23. The optical path length of each ray can be obtained by tracing this ray geometrically, as given by Eq. (30), which can be transformed into phase differences for the interfered rays using Eq. (31). The difference in the optical path lengths of each pair of interfered rays can be transformed into an intensity distribution describing the interference fringes using Eq. (32). On the other hand, the scattered rays from the outer surface of the fiber do not contribute in the interference because of the limited range of the incident rays on the fiber. This is in contrast with previous works done by Watkins [14, 15, 61]. By comparing the experimentally obtained interferograms with those reconstructed theoretically as

Figure 22.

The ray tracing diagram of an optical ray crossing a homogeneous thick optical fiber.

The relation between the position of each two interfered rays on the screen and their incidence angles on the thick fiber.

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

Figure 24.

(a) A selected and extended part of the obtained interferogram of a thick optical fiber, (b) the enhanced fringes of (a) and (c) the simulated fringes.

shown in Figure 24, Ramadan was able to determine the refractive index of the investigated thick optical fiber. The advantage is that the used system requires no matching liquid where the experiment is performed when the thick fiber is just held in air. This enables monitoring the probable variation in radius or refractive index of the fiber particularly during the manufacturing process or under external effects.

$$\Delta(\mathbf{z}) = ab + bc \cdot n\_L + cd + de \cdot n\_f + ep \tag{30}$$

$$\delta = \frac{2\pi}{\lambda} \cdot \left(\Delta(\mathbf{z}\_1) - \Delta(\mathbf{z}\_2)\right) \tag{31}$$

$$I = 4A^2 \cos^2\left(\frac{\delta}{2}\right) \tag{32}$$

where Δ(z1) and Δ(z2) are the optical path lengths of the two interfered rays. In 2004, Hamza et al. developed LFI technique in order to determine the refractive index of the core of a skin-core thick optical fiber [60]. They derived a mathematical expression for the optical paths through the fiber in order to reconstruct the interfernce pattern due to the used fiber when it is used as a thick fiber in the LFI system. By comparing the experimentally obtained patterns with the theoretically reconstructed ones, they were able to estimate the core's refractive index with an accuracy of 8 � <sup>10</sup>�<sup>4</sup> . Due to its simplicity and applicability, LFI was used, afterward, to measure the refractive index of a liquid [62] and to monitor the thickness variations of a transparent sheet inserted between the cylindrical lens and the thick fiber [63].
