1. Introduction

The fabrication of fiber soliton laser was made possible by high damage threshold of single-mode silica fiber [1]. The second harmonic generation from mode-locked and Q-switched Nd: YAG laser pulses in silica fiber was also achievable [2]. There are limits to the optical power that can be supported by optical fibers used in automotive, aerospace, electronic, and heavy manufacturing. One of these ultimate limits arises from a phenomenon called self-focusing [3, 4]. Self-focusing occurs when a beam is focused in a transparent medium, caused by the beam itself through a nonlinear process. This causes the beam radius to be reduced, allowing optical intensities to become higher leading to immediate fiber destruction. A dynamic picture of the self-focusing process for cylindrical beams was suggested [5]. The first experimental observation of self-focusing in multi-mode fibers using high-peak power pico-second pulses was reported [6]. Damage has been observed when coupling Q-switched Nd:YAG laser light into larger fibers, arising from also self-focusing effect [7, 8]. Several examples of a fiber fuse resulting in catastrophic destruction of an optical waveguide were reported elsewhere [9-13]. In silica fibers, self-focusing due to nonlinear effect requires optical power of about 10 MW/cm² [14].

In comparison to step-index fiber, the gradient-index fiber intensity profile, for the same fiber diameter, has a smaller 86% radius. The peak intensity is often about five times that of a comparable size step-index fiber. Laser material processing requires very high irradiance, thus, the gradient-index fiber may be the most appropriate choice especially when deeper penetration and high speed cutting/welding are required [15]. In contrast to the step-index fiber, GRIN fiber is preferable because the fundamental mode of the GRIN fiber is relatively more confined. Also the mode-mixing caused by bending in a GRIN fiber couples modes together while preserving radial symmetry which yields a beam with a better focus [16].

The reported works [17-24] that studied Kerr nonlinearity in optical fibers practically used in high power fiber laser delivering systems are however insufficient because none of these reports were considering GRIN fibers. Therefore, in this work we report experimental determination of the second-order index of refraction, n₂, profiles in GRIN fiber. The Fizeau interferometric study revealed the importance of waveguide dispersion profile and effect of central index dip on the nonlinear index profiles of GRIN fibers.

2. Theoretical Considerations

Understanding of nonlinear optical processes in transparent medium involves two questions: 1) how does the electromagnetic field affect the medium, and 2) how is the field affected by the medium response. The nonlinear dependence of the refractive index on the light irradiance is expressed by:

(1)

where I is the beam irradiance in units of W/m², n₀ is the linear refractive index of the material, and the coefficient of proportionality n₂ is the nonlinear refractive index. Self-focusing of laser beams in isotropic transparent solids arises from three mechanisms: electrostrictive, thermal and Kerr nonlinearity. The response times for electrostrictive and thermal self-focusing, however, are much longer than the 125-ps pulse [24]. When the optical frequency is below the electronic band gap, the self-focusing for laser beam in isotropic media due to Kerr effect can be attributed to the third-order susceptibility, χ⁽³⁾, which has a minimum response time of order of 10^-16 s. It is faster than the time resolution provided by the shortest optical pulses available today (<10 fs). Self-focusing occurs when a laser beam having a Gaussian transverse intensity distribution propagates through an optical medium whose refractive index is described by Eq. (1) considering n₂ is positive. Phenomenologically, in dielectric materials the nonlinear susceptibility is proportional to the linear susceptibility. Consequently, from Miller’s rule n is proportional to the third-order susceptibility, χ⁽³⁾ through the empirical formula [25]:

(2)

According to Vogel et al. [26], χ⁽³⁾ as a function of second-order refractive index, n₂, is given by the relation:

(3)

or

(4)

Eq. (4) reflects the fact that a high n₂ is expected for glass with high linear index of refraction. For conversion esu unit to m²/W in the SI system of units, the following formula is used:

(5)

The graded-index multi-mode optical fiber with a circular cross-section of radius r_f introduced in a silvered liquid wedge interferometer has a cladding of constant refractive index n_cl, a graded-index core of variable refractive index n_c(r) and radius r_c and a graded-index dip of variable refractive index n_d(r) and radius r_d. The fiber is immersed in a liquid of refractive index n_L close to n_cl. The equation represents the shape of multiple-beam Fizeau fringes in the dip region, i.e., for a radial distance x₁, is given by [27]:

(6)

where and α_c is a shaping parameter controlling the shape of the core index profile. Also, and α_d is a parameter controlling the shape of the dip index profile. is the fringe spacing in the liquid region and δz is the fringe shift in the fiber region, where λ is the wavelength.

3. Experimental

The experimental setup used to produce Fizeau fringes in transmission is described elsewhere [28]. A parallel beam of monochromatic light illuminates the liquid wedge interferometer. The wedge is adjusted so that the Fizeau fringes were crossing perpendicularly the fiber axis. The fringes in the liquid region were straight lines parallel to the wedge apex with an inter-spacing Δz. Immersion oil is usually used to prevent the fiber cladding from contributing a very large amount of fringe shift. The fact that light rays pass the fiber without deflection is assumed, considering only that their phases are retarded according to their optical thicknesses. Graded index multi-mode optical fibers (ITT Telecommunications Corporation, USA) with 6 dB/km average attenuation, 3.0 ns/km maximum dispersion, core diameter 50 μm and 125 μm outer diameter have been investigated.

Fig. 1 presents an interference pattern of Fizeau fringes at transmission crossing transversally a strain-free straight GRIN fiber immersed in non-matching liquid, n_L < n_cl. The straight line fringes in the liquid region as they cross the fiber cladding they follow an elliptic path. The fringe shift in the cladding represents a half ellipse of semi-principal axes r and (n_cl-n_L)r. The direction of the half ellipse is towards the wedge apex for n_L < n_cl which is the same direction of the core shift. The figure provides information about the core structure and the distinctive features of the index profile through the discontinuity in the core fringe.

Figure 1. An interferogram of Fizeau fringes at transmission crossing transversally a strain-free straight GRIN fiber immersed in non-matching liquid, nL < nclad. The fiber is perpendicular to the wedge apex. The fringe shift across the dip region is in the direction of lower refractive index

The dotted line represents the envelope of the core index profile following the basic power law with core index difference Δn_c = 0.016 at λ = 850 nm. The extension of each layer is determined from the interferogram by the projection of the fringe segment on the x-axis. The steps (in Δn = 0.0026) [29] decrease toward the fiber center while the extension of each step remains constant. The measurements showed that the extension Δr was the same for each contributing layer. The value of Δr was found to be 2.78 µm. The obtained results from the micro-interferogram show that the central dip of the fiber core has a gradient index profile with a core is made from a binary compound material such as GeO₂-SiO₂ [27]. Fibers which made of binary compound materials are designed for high bandwidth propagation using a GaAs laser [30].

4. Results and Discussion

The cladding refractive indices of the fiber samples are found to be 1.4384 ± 0 .0001, 1.4395 and 1.4568 at λ = 587.5, 486.1 and 656.3 nm, respectively. Such values mean that the cladding could be made from doped fused silica. Fig. 2 shows that at λ = 587.5 nm the graded-index core-dip index profiles of the fiber examined is a succession of step-index layers with a step-pyramid-like index profile across the fiber core. As most common, the dopant concentration is increased in equal steps from one layer to the next keeping the layer extension constant. The central part of the core index starts at 2.5 µm from the optic axis to decrease reaching its minimum value with Δn = 0.008.

Figure 2. The radial refractive index profile (at λ = 587.6 nm) of the fiber core which inclods a graded central dip

Using Eq. (4), Fig. 3 illustrates the waveguide dispersion of the calculated second-order nonlinear index radial profile of the GRIN optical fiber due to Miller [25] and Vogel et al. [26]. The nonlinear refractive index profile of the investigated graded-index multi-mode fiber is nearly parabolic following the linear refractive index behavior with an estimated error of n₂ is ± 0.1%. Taking into consideration the fundamental propagated mode, the presence of central index dip in graded-index fiber makes the comparison between GRIN fibers and multi-mode step-index fibers in the favor of GRIN fibers. GRIN fiber with central dip provides a considerable decrease in the self-focusing effect in an order of 6.5%. This structure of waveguide dispersion profile would compensate the Gaussian laser intensity radial profile which has a maximum intensity at the center of the laser beam. This means that the index dip would provide fiber-laser delivering system endure more laser intensity. In addition, it was shown that the nonlinear refractive index, n₂, of germanium doped silica glass is a linear function the GeO₂ doping [31]. Thus, by decreasing the amount of GeO₂ dopant in the inner core layers there is a possibility to design a saddle-like shape for the radial profile of n₂ which is an optimum profile shape to minimize the effect of n₂.

Figure 3. Secon-order nonlinear index profile of GRIN fiber based on linear index profile due to Miller [25] and Vogel et al. [26]

On the other hand, ignoring the fiber material dispersion has a great deal of simplistic [32]. So measuring the material dispersion profiles for the fiber glass layers may provide a further test for right prediction of the calculated nonlinear index profile. Optical glasses are usually described by their refractive indices at the helium d line (587.6 nm), n_d, and the Abbe number, ν_d, which characterizes the material dispersion. The Abbe number is given by:

(7)

The difference (n_F - n_C) represents the material dispersion where n_F and n_C are the refractive indices for reference lines of hydrogen: F (486.3 nm) and C (656.3 nm), respectively. The nonlinear refractive index (in m²/W) can be estimated from the values of n_d and ν_d by using the following Milam and Weber expression [24]:

(8)

Fig. 4 elucidates the radial change of waveguide dispersion of second-order nonlinear refractive index profile in the core and dip regions of the studied multi-mode graded-index fiber taking into consideration material dispersion. In spite of the index inversion seen in the dip region a decrease of 97.6% in the nonlinear index value in the dip region of is obtained. The waveguide dispersion of the nonlinear index profile does not follow the shape of n₂ profile which is constructed considering the linear index values (Fig. 3). To explain such behavior, the material dispersion of the fiber layers is plotted against the fiber radial distance as shown in Fig. 5. It is clear that the variation of the second-order index of refraction follows the material dispersion profile (Fig. 4) rather than the linear refractive index distribution profile (Fig. 3). This reveals that the material dispersion of the fiber layers plays the dominant role rather than linear index of refraction in the nonlinear properties of optical fibers. One of the recommendations here is to eliminate the central dip during fiber production. Also, the profile of material dispersion-base waveguide structure can be considered as a base to the proper design of (n₂) radial profile. The results are useful for production of optical fibers that can bear more laser power required for high-power ultrashort-pulse fiber laser delivering systems.

Figure 4. Secon-order nonlinear index profile of GRIN fiber based on the fiber material dispersion profile due to Milam and Weber [24]

Figure 5. Material dispersion radial profile of core and dip of GRIN fiber

5. Conclusions

GRIN fiber has a large cross-section and superior confinement structure to the laser beam. This provides high-quality focusing properties. Determination of second-order index coefficient radial profile is highly recommended to acquire more understanding of self-focusing in optical fibers. Therefore, Fizeau interferometry is applied to estimate second-order index coefficient in GRIN optical fiber. The findings pointed out that in comparison with waveguide dispersion; the material dispersion profile is a critical parameter which shapes the n₂ profile. For high power laser propagation with elimination for self-focusing effect, GRIN fiber without central dip is suggested. The results would help to design fiber having waveguide dispersion profile that provides maximum transmission capacity of high power laser systems with a good focus quality.