# Modeling of single mode optical fiber having a complicated refractive index profile by using modified scalar finite element method

## Abstract

A numerical method based on modified scalar finite element method (SC-FEM) is presented and programmed on MATLAB platform for optical fiber modeling purpose. We have estimated the dispersion graph, mode cut off condition, and group delay and waveguide dispersion for highly complicated chirped type refractive index profile fiber. The convergence study of our FEM formulation is carried out with respect to the number of division in core. It has been found that the numerical error becomes less than 2 % when the number of divisions in the core is more then 30. To predict the accurate waveguide dispersion characteristics, we need to compute expression \(\frac{{{\text{d}}^{2} \left( {\text{vb}} \right)}}{{{\text{dv}}^{2} }}\) numerically by the FEM method. For that the normalized propagation constant *b* (in terms of \(\beta\)) should be an accurate enough up to around 6 decimal points. To achieve this target, we have used 1 million sampling points in our FEM simulations. Further to validate our results we have derived the higher order polynomial expression for each case. Comparison with other methods in calculation of normalized propagation constant is found to be satisfactory. In traditional FEM analysis a spurious solution is generated because the functional does not satisfy the boundary conditions in the original waveguide problem, However in our analysis a new term that compensate the missing boundary condition has been added in the functional to eliminate the spurious solutions. Our study will be useful for the analysis of optical fiber having varying refractive index profile.

### Keywords

Chirp type’s refractive index profile Waveguide dispersion Group delay Finite element method (FEM)## 1 Introduction

In this paper, a modified finite element method (FEM) based on a variational formulation for cylindrical coordinates system, which can consider complicated refractive index profile is presented and results are simulated on MATLAB platform. Like the other numerical methods, the FEMs are widely utilized methods. The SC-FEM presented in this paper has a certain advantages over the vectorial FEM, like SC-FEM has no spurious problem. Due to this facilitation since the matrix size for our analysis is one-third to two-thirds smaller than that for traditional analysis, required memory and CPU time become very small. The result of our computation is well agreed with the previously published results by other methods like vector-FEM, finite difference method, etc.

The optical fibers can be in various structural dissimilarities just like in photonic crystal fiber has various size or shapes of holes. Many critical steps may involve during these type of optical fibers fabrication process (Zheludev 2010; Politano and Chiarello 2013; He et al. 2012; Baqir and Choudhary 2014; Li et al. 2010). The modeling process plays an important role in the development of optical fibers and related devices by evaluating the geometrical design performance such as guiding properties, mode confinement capability to mention few. Optical waveguide modeling techniques can be divided into analytical and numerical methods. Numerical methods are preferred whenever the analytical solution is not possible for certain geometry like in photonic crystal optical fiber. For the numerical methods, several approaches, this includes the scalar or vectorial finite difference method, scalar or vectorial finite element method, and Beam propagation method are preferred. Apart of that a semi-analytical method has also developed to analyze a taper optical waveguide (Politano and Chiarello 2013; Mussina et al. 2014; Walpita 1985; Rahman 1995; Raghuwanshi and Kumar 2014; Honkis 2011; Okoshi and Okamoto 1974). Instead of finite element method, finite difference method may also be preferred in certain cases due to easier formulation procedure; however accumulation of truncation error and long computation time may reduce the method feasibility. In order to overcome these problems in this paper we opted the finite element method analyses in order to produce acceptable simulation results while shorten the time. In case of weakly guiding approximation, we can efficiently use the scalar wave equation solution instead of fully vectorial method for complicated waveguides. Indeed in this paper we use the weakly guiding approximation throughout for all the cases like chirped/alpha power refractive index profile. It then obtained the scalar wave equation by ignoring the terms for the interaction between two polarized field components in the vectorial wave equations. Since in our FEM formulation we are dealing with a single mode fiber with degenerate mode (HE_{11} mode having same polarization state in principal), hence the error generated by scalar FEM while compared to vectorial FEM is negligible (Popescu 2005; Rostami and Motavali 2008; Chaudhuri and Roy 2007; Sadiku 1992; Booton 1992; Kasim et al. 2006; Gambling et al. 1977; Zhuangqi et al. 1999; Chiang 1994; Xu et al. 1993; Okamoto 2006; Sharma et al. 1981; Okamoto and Okoshi 1976; Raghuwanshi and Kumar 2012; Ghatak and Thyagarajan 1999; Hotate and Okoshi 1978; Rostami and Moyaedi 2004; Survaiya and Shevagaonkar 1999; Raghuwanshi and Talabattula 2012).

## 2 Finite element method analysis of optical fibers

_{11}mode in optical fibres having a complex refractive-index profile is described. Figure 1 reveals the core region \(0 \le r \le a\) where the refractive index can be an arbitrary profile. The maximum refractive index of the core is denoted as

*n*

_{1}(=1.5) and that in the cladding as \(n_{0} \left( { = 1.4775} \right)\). The wave equation correspond to HE

_{11}mode is given, by with

*m*= 0 as (Okoshi and Okamoto 1974; Popescu 2005; Rostami and Motavali 2008)

*E*denotes electric field. The boundary condition is given by the continuity for

*E*and

*dE*/

*dr*at \(r = a\).

*ρ*=

*D*corresponds to the core and cladding interface. Following this normalization, the wave equation and the boundary condition are rewritten as

*v*and the normalised refractive index profile

*q*(

*ρ*) are given by

*R*

_{0}−

*R*

_{N}are the field values at the sampling points to be solved and

*K*

_{0}is 0th order modified Bessel function. Also

## 3 Modeling of graded types refractive index profile of single mode optical fiber

_{11}of optical fiber while comparing with TE modes of planar slab waveguide in the \(\upalpha\)-power refractive-index profiles given by Eq. (14). We consider the total core radius to be “2a” and the sampling point in the core \({\text{N}} = 100\). The step-index slab waveguide is also analyzed by setting \(\upalpha = \infty\) in Eq. (14). Figure 2 reveals the normalized cut off frequency \({\text{V}}_{\text{c}}\). It is apparent \(\upalpha\) tends to be \(\alpha \to 100\) correspondingly \({\text{V}}_{\text{c}} \to {\text{V}}_{\text{co}} \to 2.405\). To know the accuracy of the computation in our FEM formulation, the percentage of numerical error \(\left( {{\text{V}}_{\text{c}} - {\text{V}}_{{{\text{c}}0}} } \right)/{\text{V}}_{{{\text{c}}0}} \times 100\,\left( \% \right)\) for the cut off normalized frequency in case of step index profile (where \({\text{V}}_{{{\text{c}}0}}\) is the cutoff frequency for step index profile) is computed with respect to the number of core divisions N. Figure 3 reveals the validity of our simulation by the fact that if the number of core division is around 70 the error even become less than 1 %. Figures 4, 5 and 6 shows the normalized propagation constant \(b\), field vector and propagation constant in terms of \({\beta /}{\text{k}}\) for the \({\text{HE}}_{11}\) mode of optical fiber while comparing with \({\text{TE}}_{0}\) mode of planar slab waveguides with \(\upalpha\)-power refractive-index profiles respectively. Here, we assumed \({\text{n}}_{1} = 1.5, {\text{n}}_{\text{s}} = {\text{n}}_{0} = 1.4775\) and \(r = 2a = 2\,\upmu{\text{m}}\) hence \(A = 2\,\upmu{\text{m}}\).

From the observation of the Fig. 4, it is apparent that the planar slab waveguide behaves more or less similarly as an optical fiber near cut off frequency (low *v*-number range of 0 ≤ *v* ≤ 1.3) for any type of profile. However the triangular profile shows much difference in their behvior far from cut off region. We can also conclude that the fundamental mode of an optical fiber HE_{11} behaves in a similar way as TE_{01} mode of planar slab waveguide for step index profile case.

*E*and magnetic field

*H*vectors for the case of

*HE*

_{11}is shown in Fig. 5. It is apparent from this plot that

*E*vector is perpendicular to

*H*field vector. The plot of

*E*

_{x}component for the case of

*HE*

_{11}mode is shown in Fig. 6. Figures 5 and 6 corresponds to \({\text{n}}_{1} = 1.476, {\text{n}}_{0} = 1.446\) and \(r = 3\,\upmu{\text{m}}\) (\(A = 3\,\upmu{\text{m}}\) and \(a = 1\,\upmu{\text{m}}\)) at \(\lambda = 1.55\,\upmu{\text{m}}\) while computed allowed value of \(\beta = 5.9538 \frac{1}{{\upmu{\text{m}}}}\). Once the propagation constant \(\upbeta\) is known for entire wavelength of interest to us we can predict the dispersion characteristics of an optical fiber or a waveguide having a complex refractive index profile. After knowing the normalized propagation characteristics we can calculate \(\frac{{{\text{d}}\left( {v.b} \right)}}{\text{dv}}\) and \({\text{v}}\frac{{{\text{d}}^{2} \left( {v.b} \right)}}{{{\text{dv}}^{2} }}\) for any type of refractive index profile case. Further we can calculate the waveguide dispersion by using the following expression (Okamoto 2006; Sharma et al. 1981; Okamoto and Okoshi 1976),

## 4 Modeling of linearly chirp types refractive index profile of single mode optical fiber

*r*= 0,

*α*controls the growth or decay of the profile envelope,

*N*

_{c}is the number of cycles in a core radius,

*a*is the core radius and

*n*

_{0}is the cladding refractive index. The refractive index profile can be divided into two parameters. One, the fiber parameters like \(a\), \(n_{1} ,\)\(n_{0}\) and other, the profile parameters like

*α*and

*N*

_{c}. By varying these parameters \(\left( {a, n_{0} , n_{1} , N_{c} , \alpha } \right)\), we can generate profiles from simple step index type to complex multiple cladding type as shown in Fig. 9. For an example the profile parameters are

*N*

_{c}= 0 and

*α*= 0 respectively for step index profile. Figures 10, 11 and 12 shows the normalized propagation constant

*b*, propagation constant in terms of \(\frac{\beta }{k}\) and normalized group delay \(\frac{{{\text{d}}\left( {{\text{v}}.{\text{b}}} \right)}}{\text{dV}}\) for the

*HE*

_{11}mode of optical fiber having linear chirp types of refractive-index profiles. The waveguide dispersion can be computed straight forward from the Fig. 12 and Eq. (15).

*b*and \(v\)-number. Using the described polynomial equations, we can get the similar result and more accurate response of the waveguide dispersion. The polynomial equations, which describe the relation between

*b*and

*v*-number represented in Fig. 4 can be shown by the Table 1.

The polynomial equation derived for the Fig. 4

On the basis of the polynomial mentioned in Table 1, we can obtain the same result represented in the Figs. 4, 7 and 8 as shown in the Fig. 13.

*b*versus

*v*-number relation similar to the Fig. 4. On the basis of this, we can calculate the response for \(\frac{{{\text{d}}\left( {{\text{b}}.{\text{v}}} \right)}}{\text{dv}}\), \({\text{v}}\frac{{{\text{d}}^{2} \left( {{\text{b}}.{\text{v}}} \right)}}{{{\text{dv}}^{2} }}\) represented in the Fig. 13b, c, respectively. Finally, we can calculate the waveguide dispersion, represented in the Fig. 13d. Hence, this shows that we can consider the derived polynomial as an appropriate mathematical equation for the calculation of the waveguide dispersion. In the same manner, we can show the higher order polynomial equations, which show the similar result equivalent to the Figs. 10 and 12. The Table 2 shows the derived polynomial equation.

The polynomial equation derived for the Fig. 11

## 5 Conclusion

In the conclusion of the paper, we have characterized the dispersion property of linearly chirp types of refractive index profile. The accuracy of FEM has been tested with respect to the number of core division. We have achieved very good agreement with the previously published results. It has been demonstrated that the numerical error becomes less than 2 % for the number of the core divisions in FEM analyses ≥30. Our study also reveals that the optimum chirped type of refractive index profile waveguide can be used to achieve the flat waveguide dispersion property over the band. This study will be useful in optical communication systems where low dispersion link has to be deployed. Commercial software which obeys only the certain geometrical guide-lines, instead our method is flexible to analyze for all type of geometrical shape of holes in the core for photonic crystal fiber. Moreover the accuracy of our FEM method found to be up to 6 decimal points which are sufficient enough to compute the waveguide dispersion and performance is well comparable to commercial software.

## Notes

### Acknowledgments

This work is prepared as part of the post doctorate research work done by Dr. Sanjeev Kumar Raghuwanshi under the project of Erasmus Mundus Scholarship program in a collaboration between City University London, United Kingdom and Indian School of Mines Dhanbad, Jharkhand India. This project has been Funded with support from the European Commission. This publication reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

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