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

  • Sanjeev Kumar Raghuwanshi
  • B. M. Azizur Rahman


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.


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 (HE11 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

In this section, variational formulation based on FEM analysis of the HE11 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 n1(=1.5) and that in the cladding as \(n_{0} \left( { = 1.4775} \right)\). The wave equation correspond to HE11 mode is given, by with m = 0 as (Okoshi and Okamoto 1974; Popescu 2005; Rostami and Motavali 2008)
$$\frac{1}{\text{r}}\frac{\text{d}}{\text{dr}} \left( {{\text{r}}\frac{\text{dE}}{\text{dr}}} \right) + \left( {{\text{k}}^{2} {\text{n}}^{2} -\upbeta^{2} } \right){\text{E}} = 0$$
where E denotes electric field. The boundary condition is given by the continuity for E and dE/dr at \(r = a\).
Fig. 1

Refractive-index distribution and schematic of \(\upalpha\)-power refractive index distribution of inhomogeneous optical fiber, where ‘a’ is the core radius

Before transforming Eq. (1) into a variation expression, the waveguide parameters are normalized as
$$\rho = \frac{r}{a},\quad E_{t} \left( r \right) = R\left( a \right),\quad D = \frac{A}{a}$$
where \(r = A\) and ρ = D corresponds to the core and cladding interface. Following this normalization, the wave equation and the boundary condition are rewritten as
$$\left. {\begin{array}{*{20}c} {\frac{1}{\rho }\frac{d}{d\rho } \left( {\rho \frac{dR}{d\rho }} \right) + \left[ { v^{2} q\left( \rho \right) - w^{2} } \right]R = 0,} \\ {R\left( \rho \right)\,and\frac{dR\left( \rho \right)}{d\rho }\,are\, continuos\, at\,\rho = D} \\ \end{array} } \right\}$$
Here the transverse wavenumber \(w\), the normalized frequency v and the normalised refractive index profile q(ρ) are given by
$$\left. {\begin{array}{*{20}c} {w = a^{2} \sqrt {\beta^{2} - k^{2} n_{0}^{2} } } \\ {v = ka\sqrt {\left( {n_{1}^{2} - n_{0}^{2} } \right)} } \\ {q\left( \rho \right) = \frac{{n^{2} - n_{0}^{2} }}{{n_{1}^{2} - n_{0}^{2} }}} \\ \end{array} } \right\}$$
The solution of the wave Eq. (3) under the constraint of the boundary condition can be obtained as the solution of the variational problem that makes the functional stationary: as follows (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)
$$I\left[ R \right] = - \int_{0}^{x} {\left( {\frac{dR}{d\rho }} \right)^{2} } \rho d\rho + \int_{0}^{x} {\left[ {v^{2} q\left( \rho \right) - w^{2} } \right]} R^{2} \rho d\rho$$
The validity of Eq. (5) is proved in a similar manner to that in ref (Sharma et al. 1981; Raghuwanshi and Kumar 2012). Having to solve the Eq. (5) field profile in the core is dicretised and expressed as
$$R\left( \rho \right) = \left\{ {\begin{array}{*{20}l} {\mathop \sum \limits_{i = 0}^{N} R_{i} \phi_{i} \left( \rho \right)} \hfill & {\left( {0 \le \rho \le D} \right)} \hfill \\ {\frac{{R_{n} }}{{K_{0} \left( {wd} \right)}}K_{0} \left( {w\rho } \right)} \hfill & {\left( {\rho > D} \right)} \hfill \\ \end{array} } \right.$$
where R0 − RN are the field values at the sampling points to be solved and K0 is 0th order modified Bessel function. Also
$$\left. {\begin{array}{*{20}c} {{\text{R}}_{\text{i}} = {\text{R}}\left( {\uprho_{\text{i}} } \right)\quad {\text{for}}\, \left( {{\text{i}} = 0 - {\text{N}}} \right) } \\ {\uprho_{\text{i}} = {\text{i}}\frac{\text{D}}{\text{N}}\quad {\text{for}}\, \left( {{\text{i}} = 0 - {\text{N}}} \right) } \\ \end{array} } \right\},$$
The solutions in the cladding and the substrate are given by the analytical functions. The sampling function \(\upphi_{\text{i}} \left(\uprho \right)\) becomes unity at \(\uprho =\uprho_{\text{i}}\), becomes zero at the neighboring sampling points \(\uprho =\uprho_{{{\text{i}} - 1}}\) and \(\uprho =\uprho_{{{\text{i}} + 1}}\), and is zero throughout all other regions. The precise expressions of \(\upphi_{\text{i}} \left(\uprho \right)\) are given by,
$$\upphi_{0} \left(\uprho \right) = \left\{ {\begin{array}{*{20}l} {\frac{\text{N}}{\text{D}}\left( {\uprho_{\text{i}} -\uprho} \right)} \hfill & {0 \le\uprho \le\uprho_{\text{i}} } \hfill \\ 0 \hfill & {\text{All other areas}} \hfill \\ \end{array} } \right.$$
$$\upphi_{\text{i}} \left(\uprho \right) = \left\{ {\begin{array}{*{20}l} {\frac{\text{N}}{\text{D}}\left( {\uprho -\uprho_{{{\text{i}} - 1}} } \right)} \hfill & {\left( {\uprho_{{{\text{i}} - 1}} \le\uprho \le\uprho_{\text{i}} } \right)} \hfill \\ {\frac{\text{N}}{\text{D}}\left( {\uprho_{{{\text{i + }}1}} -\uprho} \right)} \hfill & {\left( {\uprho_{\text{i}} \le\uprho \le\uprho_{\text{i + 1}} } \right)} \hfill \\ 0 \hfill & {{\text{All}}\,{\text{other}}\,{\text{areas}}} \hfill \\ \end{array} } \right.$$
$$\upphi_{\text{N}} \left(\uprho \right) = \left\{ {\begin{array}{*{20}l} {\frac{\text{N}}{\text{D}}\left( {\uprho -\uprho_{\text{N - 1}} } \right)} \hfill & {\left( {\uprho_{{{\text{N}} - 1}} \le\uprho \le\uprho_{\text{N}} } \right)} \hfill \\ 0 \hfill & {\text{All other areas}} \hfill \\ \end{array} } \right.$$
Since the sampling function here is a linear function of \(\uprho\), Eq. (6) means that the continuous function \({\text{R}}\left(\uprho \right)\) is approximated by the broken lines. The normalized refractive index distribution \({\text{q}}\left(\uprho \right)\) is also approximated, by using the sampling function, as
$${\text{q}}\left(\uprho \right) = \mathop \sum \limits_{{{\text{i}} = 0}}^{\text{N}} {\text{q}}_{\text{i}}\upphi_{\text{i}} \left(\uprho \right)$$
$${\text{q}}_{\text{i}} = \frac{{{\text{n}}^{2} \left( {\uprho_{\text{i}} } \right) - {\text{n}}_{\text{s}}^{2} }}{{{\text{n}}_{1}^{2} - {\text{n}}_{\text{s}}^{2} }}$$
Substituting Eq. (6) in Eq (5), we obtain the functional
$$I = - \int_{0}^{D} {\left( {\frac{dR}{d\rho }} \right)^{2} } \rho d\rho + \int_{0}^{D} {\left[ {v^{2} q\left( \rho \right) - w^{2} } \right]} R^{2} \rho d\rho - \frac{{wDK_{1} \left( {wD} \right)}}{{K_{0} \left( {wD} \right)}}R_{N}^{2}$$
The detail derivation of above Eq. (13) is given in Appendix “Derivation of Eq. (13)”.

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

Once the eigenvalue equation for matrix \({\text{C}}\) as described in Appendix “Derivation of eigenvalue equation using stationary condition” is solved to find the allowed values of wave propagation constant \(\upbeta\) leading to obtain the corresponding eigenvector \({\text{R}}_{0} , {\text{R}}_{1} , \ldots {\text{R}}_{\text{N}}\) by simple matrix operation. Here \({\text{R}}_{\text{i}} \left( {{\text{i}} = 1 - {\text{N}}} \right)\) is obtained with respect to \({\text{R}}_{0} \left( { \ne 0} \right)\), while \({\text{R}}_{0}\) is still yet to find out. The total optical power \({\text{P}}\) has to be normalized to obtain \({\text{R}}_{0}\). Figure 1, show the \(\upalpha\)-power refractive index profiles given by (Okamoto 2006; Okamoto and Okoshi 1976), for an optical fiber and planar slab waveguide respectively
$$\left. {\begin{array}{*{20}c} {n^{2} \left( r \right) = \left\{ {\begin{array}{*{20}l} {n_{1}^{2} - \left( {n_{1}^{2} - n_{0}^{2} } \right)\left( {\frac{r}{a}} \right)^{\alpha } ,} \hfill & {\left( {0 \le r \le a} \right)} \hfill & {for\,Fiber} \hfill \\ {n_{0}^{2} } \hfill & {\left( {r > a} \right)} \hfill & {} \hfill \\ \end{array} } \right.} \\ {{\text{n}}^{2} \left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}l} {{\text{n}}_{\text{s}}^{2} } \hfill & {\left( {{\text{x}} < 0} \right)} \hfill & {} \hfill \\ {{\text{n}}_{1}^{2} - \left( {{\text{n}}_{1}^{2} - {\text{n}}_{\text{s}}^{2} } \right)\left| {\frac{\text{x}}{\text{a}} - 1} \right|^{\upalpha} } \hfill & {\left( {0 \le {\text{x}} \le 2{\text{a}}} \right)} \hfill & {\text{for planar slab}} \hfill \\ {{\text{n}}_{\text{s}}^{2} } \hfill & {\left( {{\text{x}} > 2{\text{a}}} \right)} \hfill & {} \hfill \\ \end{array} } \right.} \\ \end{array} } \right\}$$
The planar slab waveguide is a basic structure through one can design various types of integrated optical waveguide structure. In this paper the objective is to compare the circular core fiber case with planar slab waveguide case. This is because the coordinates are different but the wave equation is same for both the cases (Raghuwanshi and Rahman 2015a, b; Raghuwanshi et al. 2014). Hence the validity of the results can be established better. Next, we shows the results of FEM analyses for HE11 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 \(A = 2\,\upmu{\text{m}}\).
Fig. 2

Cutoff normalized frequency \({\text{V}}_{\text{c}}\) of optical fiber for HE11 mode with \(\upalpha\)-power refractive-index profiles

Fig. 3

Percentage error for the cut-off normalized frequency with respect to the number of core divisions \({\text{N}}\) in FEM analysis for step index profile for HE11 mode

Fig. 4

Normalized propagation constant (b) of optical fiber (HE11 mode) and planar slab (TE01 mode) waveguides with \(\upalpha\)-power refractive-index profiles

Fig. 5

Cross sectional view of the a electric field and b magnetic field vectors for the case of \(HE_{11} \left( {\lambda = 1.55 \,\upmu{\text{m}}} \right)\) mode

Fig. 6

Plot of electric field vector Ex component for the case of \(HE_{11} \left( { \lambda = 1.55 \,\upmu{\text{m}}} \right)\) mode

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 HE11 behaves in a similar way as TE01 mode of planar slab waveguide for step index profile case.

Figure 4 also reveals that in case of triangular profile case the power of fundamental mode is more tightly confined in the core region of optical fiber hence these properties can be used for strongly guided fiber applications. Cross sectional view of the electric field E and magnetic field H vectors for the case of HE11 is shown in Fig. 5. It is apparent from this plot that E vector is perpendicular to H field vector. The plot of Ex component for the case of HE11 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),
$${\text{D}}_{\text{W}} = - \frac{{\left( {{\text{n}}_{1} - {\text{n}}_{\text{s}} } \right)}}{{{\text{c}}\uplambda}}{\text{v}}\frac{{{\text{d}}^{2} \left( {v.b} \right)}}{{{\text{dv}}^{2} }}$$
Figures 7 and 8 reveals the normalized delay \(\frac{{{\text{d}}\left( {{\text{v}}.{\text{b}}} \right)}}{\text{dv}}\) and the normalized waveguide dispersion \({\text{v}}\frac{{{\text{d}}^{2} \left( {{\text{v}}.{\text{b}}} \right)}}{{{\text{dV}}^{2} }}\) respectively computed by FEM method. The ripples in these figures are due to a limited computational capability of double derivatives terms in waveguide dispersion characteristics. To better represent these graphs we have adopted the higher order polynomial expression scheme which is discussed in next section. To compute the waveguide dispersion the value of normalized propagation constant should be an accurate enough up to at least 6 decimal points.
Fig. 7

Normalized delay \(\frac{{{\text{d}}\left( {\text{Vb}} \right)}}{\text{dV}}\) of optical fiber with \(\upalpha\)-power refractive index profiles

Fig. 8

Normalized waveguide dispersion parameter \({\text{v}}\frac{{{\text{d}}^{2} \left( {\text{Vb}} \right)}}{{{\text{dV}}^{2} }}\) of optical fiber with \(\upalpha\)-power refractive index profiles

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

In this section we will apply our finite element formulation further to estimate the waveguide dispersion for chirped refractive index profile. Now here we define a variety of more complicated than previous case a generalized linear chirp type refractive index profile and is defined as (Raghuwanshi and Talabattula 2012; Raghuwanshi and Rahman 2015a, b),
$${\text{n}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}l} {{\text{n}}_{1} - \left( {{\text{n}}_{1} - {\text{n}}_{0} } \right)\left\{ {1 + \left|\frac{r}{\alpha }\right| - 1} \right\}\exp \left( { - \alpha \left|\frac{r}{ - \alpha }\right|} \right)\cos \left\{ {2\uppi{\text{N}}_{\text{c}} \left( {\frac{r}{\alpha }} \right)^{2} } \right\}} \hfill & {{\text{r}} \le {\text{a}}} \hfill \\ {{\text{n}}_{1} \left( {1 -\Delta } \right)} \hfill & {{\text{r}} > {\text{a}}} \hfill \\ \end{array} } \right.$$
where \({\text{n}}_{1}\) is the refractive index at the center of the waveguide at r = 0, α controls the growth or decay of the profile envelope, Nc is the number of cycles in a core radius, a is the core radius and n0 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 Nc. 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 Nc = 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 HE11 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).
Fig. 9

Various kinds of chirp type refractive index profiles produce by controlling the refractive index profile parameters

Fig. 10

Normalized propagation constant (b) of optical fiber with linear chirp types of refractive-index profiles

Fig. 11

Propagation constant in terms of \(\frac{\beta }{k}\) as a function of v for optical fiber with linear chirp types of refractive index profiles

Fig. 12

Normalized delay \(\frac{{{\text{d}}\left( {\text{Vb}} \right)}}{\text{dV}}\) of optical fiber with linear chirp types of refractive index profile

It is apparent from the Figs. 10, 11, 12 and 13 that the waveguide dispersion can be substantially reduced, when we deployed the optical waveguide having a complicated refractive index profile. The ripples in the Fig. 12 are due to the limited accuracy of FEM simulation. However this problem can be solved by deriving the appropriate higher order polynomial for Figs. 10 and 12. In order to obtain the more accurate results we have derived the higher order polynomial equations, which established the relationship mathematically between 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.
Fig. 13

Simulation result obtained by the derived polynomial represented in the Table 1: ab versus v-number relation, b\(\frac{{d\left( {b.v} \right)}}{dv}\,{\text{versus}}\,v\)-number relation, c\(v\frac{{d^{2} \left( {b.v} \right)}}{{d{\text{v}}}}\,{\text{versus}}\,v\)-number relation, d waveguide dispersion for different value of \(\alpha\)

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.

Figure 13 shows that, we are able to obtain the similar results using the derived polynomials represented in the Table 1. Figure 13a shows the 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.
Table 2

The polynomial equation derived for the Fig. 11

On the basis of the derived polynomial equations the waveguide dispersion plot can be represented as shown in the Fig. 14d. It is apparent while comparing Fig. 13d with Fig. 14d that in case of complicated chirped type profile case, the waveguide dispersion is flat and negative. Earlier also the profile as shown in Fig. 9c, has been analyzed in greater detail for the case of circular core fiber (Raghuwanshi and Kumar 2012). In literature (Raghuwanshi and Kumar 2012), it has been shown; \(\pm 1.5\,{\text{ps}}/{\text{km}} - {\text{nm}}\) waveguide dispersion over the band, but in our case the waveguide dispersion is even smaller as well as flat over the band as reveal in Fig. 14d. From the Fig. 14d, it is also apparent that the optimum parameter for lower waveguide dispersion is \(N_{c} = 2.5, \alpha = 1.5\), corresponds to Fig. 9c.
Fig. 14

Simulation result obtained by the derived polynomial represented in the Table 2: ab versus v-number relation, b\(\frac{{d\left( {b.v} \right)}}{dv} {\text{versus }}v\)-number relation, c\(v\frac{{d^{2} \left( {bv} \right)}}{{dv^{2} }}{\text{versus}} v\)-number, d waveguide dispersion for different value of \(N_{c} , \alpha\)

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.



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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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sanjeev Kumar Raghuwanshi
    • 1
  • B. M. Azizur Rahman
    • 1
  1. 1.Instrumentation and Sensor Division, School of Engineering and Mathematical SciencesCity University LondonLondonUK

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