In new generation of optical communication networks, simultaneous transmission of several information channels has become possible over a single optical fiber. But dispersion and dispersion slope of optical fibers are important factors, which limit high capacity optical fiber communication systems to perform high speed transmission.

In this paper, silica core photonic crystal fibers (PCFs) with three and six air–hole rings in the cladding are investigated for effective refractive index, propagation constant and dispersion characteristics, using FDTD method. It is shown that by increasing the number of the air–hole rings in the cladding around the core, the negative dispersion of the PCFs will increase, too. The obtained negative dispersion of –300 ps/nm.km is comparatively high with respect to the published results in the related literature.

Keywords: dispersion, modeling, FDTD method, photonic crystal fibers

Photonic crystal fibers (PCFs) have attained popularity in designing of compact optical devices utilized in optical networks^1,2 and optical sensing systems.^3,4 The widely used applications are in designing of dispersion compensators commonly used in WDM optical networks reducing the number of required repeaters.^5,6 One of the comparative characteristic features of the PCFs is their flexibilities in response to dispersion effects on propagating pulses present in a long–haul transmission optical fiber line.^7,8

Till date, the photonic crystal fibers (PCFs) are known to be new types of optical fibers with different structures and profiles as compared to conventional optical fibers that are used in optical communication systems.^9,10 Having different optical characteristics, the PCFs as optical devices have been used in various applications in optical systems, such as super continuum spectrum generation,¹¹ high optical power transfer,¹² dispersion compensations,¹³ tunable optical filters,¹⁴ Erbium–doped optical amplifiers,¹⁵ Bragg grating–based devices,¹⁶ optical nonlinear applications.^17,18

Different methods are used to evaluate dispersion properties of the PCFs with various structures.¹⁹ In this paper, by using FDTD analysis, two different PCF structures are investigated for utilization in compensations of positive dispersion in optical fiber as a transmission link.

For determination of optimal parameters values of the PCF, analytical and numerical methods, such as the scalar effective index method,²⁰ the vectorial effective index method (VEIM),²¹ the improved vectorial effective index method (IVEIM),²² and the finite difference frequency domain (FDFD) method,^23,24 are usually utilized. Couple of reports on designs of dispersion compensating fibers (DCFs) are published on optimization of compensation of the dispersions using PCFs in optical transmission bands ranging from 1460 nm to 1675 nm.^25,26

A survey shows a design of a pentagonal PCF with large flattened negative dispersion by using the full vector finite element method (FEM) where the average negative dispersions for two optimized designs were −611.9 ps/nm/km over 1,460–1,625 nm and −474 ps/nm.km over 1425–1675 nm wavelength bands, respectively.²⁷ In another report a porous–core circular PCF with circular arrangement of air holes, both in the periodic cladding and the porous core is simulated by using an efficient FEM. It has shown a flattened dispersion of $\pm$ 0.09 ps/THz/cm of 0.9–1.3 THz.²⁸

An ultra–flattened dispersion over the range of –210 – 15 ps/nm.km is reported in²⁹ by doping the core of a PCF with Germanium. In another report, a design of a hybrid PCF with elliptical and circular air holes of hexagonal layout with triangular lattice of five rings around the solid core is presented in,³⁰ where they obtained large flattened dispersion is of the order of 4.88 ps/nm.km over a wavelength range of 1200–1800 nm.

In our previous report, by using two–dimensional finite difference time domain (2D–FDTD), three PCF structures were proposed to optimize the dispersions by considering the effects of the geometrical parameters, such as air–hole diameters and the center–to–center spacing between the holes (pitch). The air–holes were arranged in the cladding in the form of triangular lattice by constituting 11 rings around the core.^31,32

For a linear, isotropic, non–dispersive material with no source, the time–variant Maxwell’s equations are expressed as follows:^9,33,34

$\frac{\partial \overrightarrow{H}}{\partial t}=-\frac{1}{\mu (r)}\nabla \times \overrightarrow{E}$                 (1)

$\frac{\partial \overrightarrow{E}}{\partial t}=\frac{1}{\epsilon (r)}\nabla \times \overrightarrow{H}-\frac{\sigma (r)}{\epsilon (r)}\overrightarrow{E}$   (2)

Where$\epsilon (r),\mu (r),\sigma (r)$ are susceptibility, permeability, and conductivity of the dielectric material, respectively, and$\overrightarrow{E}$ and$\overrightarrow{H}$ are respective electric and magnetic fields vectors. Using Yee cell approach, the Maxwell’s equations can be written in the form of partial differential equations with respect to time and space.³⁵

If we indicate$\text{β}$ as the propagation constant along$z$ direction in a PCF, then the expression $\phi (x,y,z)=\phi (x,y)e^{j\beta z}$  will denote component of each field where$j^2=-1$ . Therefore, the differentiation of Maxwell’s equations with respect to$z$ may be replaced by$j\beta$ and the fields would be expressed only in terms of transverse components.

A 2D Yee unit cell across the fiber cross section is shown in Figure 1.³⁵ The $x-$ component of first curl of the Maxwell’s equation for magnetic field is derived as:³³

$\frac{\partial H_x}{\partial t}=-\frac{1}{\mu }\left(\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}\right)$      (3)

Figure 1 2D yee unit cell across the fiber cross section.³⁵

By discretization of Equation (3) in terms of time and space and by using Yee cell technique, we obtain:

$\overrightarrow{H_x}|{}_{i,k}^{n+1/2}=\overrightarrow{H_x}|{}_{i,k}^{n-1/2}-\frac{\Delta t}{{\mu }_{i,k}}\left(\frac{\overrightarrow{E_z}|{}_{i,k+1}^n-\overrightarrow{E_z}|{}_{i,k}^n}{\Delta y}-j\beta \overrightarrow{E_y}|{}_{i,k}^n\right)$ (4)

Where$n$ is the discrete time step,$i$ and$k$ denote grid points of discretization in$xy-$ plane, and $\Delta t$ , $\Delta x$ , and $\Delta y$  denote the time increment, distances between two adjacent points along$x$ and$y$ directions, respectively. Similarly, the components of the other fields can also be derived.

To reduce the computational time, one prefers to consider the real numbers in above equation. For this particular reason, let us consider the real parts of $E_z,H_x,\text{and}{H}_y$  to be$\cos (\text{β}z+\text{φ})$ and that of$H_z,E_x,\text{and}{E}_y$ to be$\sin (\text{β}z+\text{φ})$ . Therefore, Equation (4) reduces to:³⁵

$H_x|{}_{i,k}^{n+1/2}=H_x|{}_{i,k}^{n-1/2}-\frac{\Delta t}{{\mu }_{i,k}}\left(\frac{E_z|{}_{i,k+1}^n-E_z|{}_{i,k}^n}{\Delta y}-\beta E_y|{}_{i,k}^n\right)$     (5)

The$y$ and$z$ components of magnetic fields are obtained as follows:

$H_y|{}_{i,k}^{n+1/2}=H_y|{}_{i,k}^{n-1/2}+\frac{\Delta t}{{\mu }_{i,k}}\left(\frac{E_z|{}_{i+1,k}^n-E_z|{}_{i,k}^n}{\Delta x}-\beta E_x|{}_{i,k}^n\right)$    (6)

$H_z|{}_{i,k}^{n+1/2}=H_z|{}_{i,k}^{n-1/2}+\frac{\Delta t}{{\mu }_{i,k}}\left(\frac{E_x|{}_{i,k+1}^n-E_x|{}_{i,k}^n}{\Delta y}-\frac{E_y|{}_{i+1,k}^n-E_y|{}_{i,k}^n}{\Delta x}\right)$    (7)

Similarly, one can obtain the components of electrical fields. The stability of the FDTD numerical method is determined by the following expression:³⁵

$\Delta t\le \frac{1}{c\sqrt{\Delta x^{-2}+\Delta y^{-2}+{(\beta /2)}^{-2}}}$             (8)

For our boundary computational treatment, we use perfectly matched layers method. One of the key characteristic parameters of PCF is chromatic dispersion, which is summation of waveguide and material dispersions. The expression for total dispersion, consisting of material and waveguide dispersion is given as:

$D_{tot}=-\frac{\lambda }{2\pi c}\left(2\frac{d\beta }{d\lambda }+\lambda \frac{d^2\beta }{d{\lambda }^2}\right)$                (9)

To simplify the above equation, we define normalized propagation constant${\beta }_{\text{N}}\text{(=}\beta \text{/}{\text{k}}_{\text{0}}\text{=2}\pi \beta \text{/}\lambda \text{)}$ and substitute it in (9) to obtain:^9,18,10

$D_{tot}=-\frac{\lambda }{c}\left(\frac{d^2{\beta }_{\text{N}}}{d{\lambda }^2}\right)\pi$             (10)

Where$\lambda$ and$c$ are wavelength and velocity of light in a vacuum, respectively. The effective refractive index of fundamental mode is defined as $n_{eff}=\beta \text{(}\lambda /2\pi \text{)}$ .³⁶

To evaluate Equation (10) in terms of${\beta }_N$ ,$d{\beta }_{\text{N}}/d\lambda$ ,$d^2{\beta }_{\text{N}}/d{\lambda }^2$ , we use FDTD method. The refractive index of the core material is determined by using Sellmeier formula as:^18,37

$n_{silica}^2(\lambda )=1+{\displaystyle \sum _{k=1}^3\frac{a_k{\lambda }^2}{{\lambda }^2-b_k^2}}$     (11)

Where$\lambda$ is the wavelength of light, and$a_k$ and$b_k$ are the Sellmeier coefficients. The practical values of $a_k$  and $b_k$  for $k=1,2,3$  are given as:³⁸

$a_1=0.6961663,a_2=0.4079426,a_3=0.8974794,$

$b_1=0.0684043,b_2=0.1162414,b_3=\mathrm{9.896161.}$

In this paper, by using the above equations and other related expressions in FDTD method, two PCF samples with different numbers of air–hole rings are selected for design in order to study their dispersion behaviors.

The selected PCF structure with air–hole of equal diameters$d=0.75\mu \text{m}$ and air–hole spacing of$\Lambda =2\mu \text{m}$ arranged in three rings in the form of triangular lattice is illustrated in Figure 2. The core diameter is taken as $1.45\mu \text{m}$ .

The spectrum of chromatic dispersion of fundamental mode in the wavelength range of 1100–2000 nm is illustrated in Figure 3 (Red curve). The dispersion at 1510 nm is obtained as –140 ps/nm.km. The dispersion of the proposed PCF in this case is nearly twice the dispersion of conventional dispersion compensating fiber (i.e.,–80 ps/nm.km).

Figure 2 The designed PCF with three rings of air–holes.

Figure 3 Dispersion spectra of the PCFs with three and six circular air–hole rings in the cladding region.

As of previous analysis, a PCF structure is considered with the same air–hole diameter of$d=0.75\mu \text{m}$ , core diameter of $1.45\mu \text{m}$ and a different air–hole spacing of$\Lambda =1.1\mu \text{m}$ . In this case, six air–hole rings are considered in the cladding region (in this case, figure not shown).³⁶

By using the home–made programming code, in this case as well, the chromatic dispersion spectrum has been depicted in Figure 3 (Blue curve). In this case, the dispersion value of –300 ps/nm.km is obtained, which twice the value is obtained in the previous design of the PCF with three circular air–hole rings. The results obtained here is comparable to the ones reported,¹⁸ in which the proposed structure with seven air–hole rings were used. The reported dispersion value was –150 ps/nm.km.

In the simulations performed by FDTD method, by varying$\beta$ values in each case, the resonance fundamental frequencies, wavelengths, and effective refractive indices are determined and tabulated in Table 1.^17,18

In this paper, by using FDTD method, two proposed photonic crystal fibers have been analyzed with three and six air–hole rings in the cladding arranged in triangular lattices. In the case of three air–hole rings in the cladding of the PCF, for air–hole with equal diameters of $d=0.75\mu \text{m}$ and air–hole spacing of$\Lambda =2\mu \text{m}$ , the dispersion value is determined to be –140ps/nm.km at wavelength 1510nm.

In case of the PCF with six air–hole rings, air–hole spacing of$\Lambda =1.1\mu \text{m}$ , and with the same air–hole of diameter of$d=0.75\mu \text{m}$ , the dispersion value is found to be –300ps/nm.km. In both the cases, the diameters of the PCF core are taken as$1.45\mu \text{m}$ .

It is further shown that when the air–hole spacing$\Lambda$ is decreased by a ratio of 55% and by doubling the number of air–hole rings, the dispersion will increase more than twice the value. The PCF with high negative dispersion value is suitable for design of high dispersion compensating fiber used in long–haul optical transmission links.

The authors greatly acknowledge the allotment of the post–graduate student by GJK Institute of Higher Education, Qazvin, Iran, for collaboration in the running academic project of optical group at ITRC.

Authors declare there is no conflict of interest.

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