Photonic crystal fiber (PCF) is a new type of micro-structured optical fiber that consists of small, closely packed air holes. It is produced using the stack-and-draw method. In the PCF, light is confined inside the silica tubes. Special properties of PCF have attracted a number of researchers in recent years. [1-3] introduced a new type of fiber to carry the light signal in a hollow core by means of photonic band-gap cladding. [4] proposed another type of PCF with a cladding structure with periodic air holes in silica. The main parameters considered in designing a PCF are the diameter d of air holes and the distance between the centers of two adjacent air holes D, which is shown in Fig. 1. The PCF coupler is one of the key elements in advanced optical networks [5-9].

Compared to conventional optical couplers, PCF provides significant features such as flexible dispersion, high nonlinearity, and support for single-mode transmission. Realization of dual-core PCF allows us to design efficient PCF couplers, wavelength multiplexers, de-multiplexers, splitters, filters, and sensors [10-14]. Dual-core PCF couplers have numerous advantages over conventional optical couplers and are more flexible to design, easy to make, and present a shorter coupling length. The important physical properties of dual-core PCF are the coupling length and birefringence [15-19].

The main objective of this work is to analyze the physical properties of the dual-core hexagonal-shaped PCF to achieve high birefringence and short coupling length for different values of structural parameters, and that the proposed structure is designed to be implemented as a polarization splitter.

The physical structure of dual-core PCF is shown in Fig. 1. The important structural parameters of our proposed structure are d the diameter of the air holes, and the pitch D, which is the distance between two adjacent air holes in the cladding. In our proposed structure, d is 1.4 µm, while D is 2.2 µm. The refractive index of air hole n_d = 1. Here silica is used as a background material with a refractive index of n_si = 1.45. A dual-core hexagonal-shaped PCF is designed by eliminating the two air holes on both sides of the middle air hole in parallel lines. Hence, asymmetry is introduced in the structure, which creates high birefringence and short coupling length. The proposed structure is designed to be implemented as a polarization splitter. Different properties of PCF are analyzed by changing the value of d in the air holes in the cladding for d/D values of 0.5, 0.6, 0.7, 0.8, and 0.9. The properties are analyzed using a solid core of silica material. The proposed polarization splitter’s structure is shown in Fig. 2.

The dual-core hexagonal PCF structure is designed by varying the structural parameters D and d and simulated using the finite-element method (FEM). Figure 3 shows the simulation results for the proposed PCF structure. It shows the light confinement in the x- polarized direction at a wavelength of 1.55 µm. Also, it shows the four different supermodes, and explains the field distribution in the fiber core.

Maxwell’s equations are used to study electromagnetic wave propagation in the photonic crystal structure. Maxwell’s third and fourth equations in the magnetic field form is given by

where ω is the angular frequency, c is the speed of light in vacuum, and ε(r) is the relative permittivity of the material.

Let us assume that the two-dimensional photonic crystal system is periodic in the x and y directions and homo-geneous in the z direction. The modes in photonic crystals are categorized into two polarizations, as the TM mode and TE mode. For the TE mode the magnetic field is along the z direction , and for the TM mode the electric field along the z direction .

To calculate the band gap, quadratic eigenvalue calculations are made over a unit cell of the photonic crystal lattice with Floquet periodic boundary conditions. We optimize the structural parameters of the PCFs and use the plane-wave method to calculate the band gap of the band-gap-guiding PCFs, and FEM to model all of the properties. The band gap, index-guiding mode line, the x and y-polarized guiding mode lines, and the cladding line are shown in Fig. 4 for the values of n_d = 1, n_si = 1.45, d = 1.4 µm and D = 2.2 µm.

The refractive index of the proposed structure in PCFs exhibits a strong wavelength dependence, very different from pure silica, which allows PCFs to be designed with a new set of features unattainable within the classical approach. Figure 5 shows that when the wavelength is increased, the value of the effective index decreases and is shown in quasi-TE for the even supermode. Effective index for different d/D decreases as wavelength increases. For smaller wavelengths, the light confinement is strong and the effective index value is high, but at higher wavelengths the effective index value decreases. At 1.55 µm, the effective index is reduced as the d/D ratio is increases. Table 1 shows the effective index for different d/D at 1.55 µm.

Birefringence is defined as the difference between the propagation constants or mode indices of the slow and fast polarization modes. Birefringence is the difference between the mode indices of the orthogonally polarized modes:

where and are effective indices in the x-polarized and y-polarized directions respectively. From Fig. 6, it is observed that the birefringence increases at higher wavelengths when the size of each air hole increases. Hence, at a wavelength of 1.55 µm the birefringence increases as the d/D value is increased. High birefringence is obtained due to the asymmetric structure. Birefringence curves are shown for the even supermode.

For all d/D values, the birefringence is found to be of the order of 10⁻³ at 1.55 µm. Table 2 shows the obtained birefringence for different d/D values at 1.55 µm.

In accordance with mode-coupling theory, the even and odd supermodes describe the coupling length of a dual-core fiber. Modes of an individual core having symmetric field distribution make even supermodes, and modes of an individual core having asymmetric field distribution make odd super modes. The dual-core PCF coupling length is defined as

where and are the p-polarized even and odd supermodes’ effective indices. and are the propagation constants of p-polarized even and odd supermodes. The fiber’s coupling length depends on the structure of the PCF. Equation 3 explains the coupling-length calculation for x- and y-polarized light. The dual-core hexagonal-shaped PCF structure has a better coupling length for larger d/D values. When the wavelength increases, the proposed dual-core hexagonal-shaped PCF gives a small coupling-length. If the length of the coupling is very small, then a small length coupler or splitter can be realized. Figs. 7 and 8 show the variation of coupling length with wavelength, for x- and y-polarized light respectively. Comparing Figs. 7 and 8, the y-polarized light’s coupling length is much less than the coupling length of x-polarized light. The birefringence is very small for smaller air holes, but the mode coupling is very strong (small coupling length). Large-diameter air holes give small mode coupling and higher birefringence. Hence, the d/D value of 0.7 is selected as the best dual-core hexagonal-shaped PCF structure for splitter realization. The coupling length for x- and y-polarized light at 1.55 µm is listed in Table 3.

A dual-core hexagonal-shaped PCF based polarization splitter is implemented. From the simulation results, it is concluded that, the value d/D = 0.7 is the optimized structure. It has higher birefringence and very short coupling length. The proposed PCF structure yield the high value of 1.802 × 10⁻³ for birefringence and 1.0962 mm as coupling length for x-polarized light. A coupling length of 0.6974 mm is obtained for y-polarized light. The results show that the coupling length of the y-polarized light is very much less than that for the x-polarized light. Different coupling lengths are obtained for x- and y-polarized light, because of the high birefringence.

In this work, two solid cores are guiding the light by an index-guiding mechanism. When broadband light is applied at the input, the linearly polarized light beams are transferred through the fiber. Assume that the power transmitted in cores C₁ and C₂ at the input side have values of 1 and 0 respectively. As per mode-coupling theory, the light will be completely transferred from one core to another core at the coupling length.

Higher birefringence is required for separating the input into x- and y-polarized light. In the simulation results, y-polarized light gives a lesser coupling length. Therefore, y-polarized light will be completely transferred into the other core. Figs. 9 and 10 illustrate the power transfer of x- and y-polarized light at a wavelength of 1550 nm, for a 3 mm fiber.

Normalized power transmission of x-polarized light in cores C₁ and C₂ is illustrated in Fig. 11. First, the light source is connected to core C₁, but the x-polarized light is switched to core C₂ at the coupling length of 1.0962 mm. At 1.0962 mm coupling length, it is observed that power transmission in core C₁ is very low and power transmission in core C₂ is large. Again, with the similar length, power transmission of x-polarized light will be greater in core C₁ and very much less in core C₂.

Normalized power transmission of y-polarized light in cores C₁ and C₂ is shown in Fig. 12, from which it is observed that y-polarized light will completely transfer to core C₂ at a coupling length of 0.6974 mm. Again, with the similar distance, it changes to core C₁. At a fiber length of 1.9 mm, the maximum amount of y-polarized light is directed to core C₂, while x-polarized light is coupled to core C₁. Therefore, for a 1.9 mm fiber the x- and y-polarized light can be separated, as shown in Figs. 12 and 13 respectively, as long as the length of the PCFs

where a and b are positive integers. For a coupling device, the values of a and b must be the same (either odd or even numbers), whereas for a splitter they must be different.

At 1.9 mm, the maximum amount of x-polarized light is transferred in core C₁ and the y-polarized light is transferred in core C₂. Therefore, at a fiber length of 1.9 mm, the connected light source can be separated as x- and y-polarized light.

The efficiency of the proposed polarization splitter is measured with the help of the extinction ratio (E_r). For core C₁, the splitting ratio is defined as

This splitting ratio indicates the performance of the proposed splitter. The splitting ratio is calculated using Equation. 4 and is shown in Fig. 13. A polarization splitter is realized for 1.9 mm length using the optimized structure with d/D ratio 0.7. The result shows that the extinction ratio of this dual-core hexagonal-shaped PCF splitter can give −34.988 dB at 1550 nm wavelength, which is better than −20 dB at wavelengths from 1508 -to- 1589 nm. Its bandwidth in one band gap is about 81 nm. The proposed splitter gives an improved extinction ratio, compared to existing splitters [20-24]. The comparisons are shown in Table 4.

In this paper, a dual-core hexagonal-shaped PCF was designed. In addition, its physical properties were analyzed to obtain high birefringence and short coupling length for different values of structural parameters, and were analyzed for a polarization-splitter application. The optimized structure with d/D of 0.7 yields a birefringence of 1.802 × 10⁻³ and a 0.6974 mm coupling length. A 1.9 mm long fiber polarization splitter is realized with a splitting ratio of −34.988 dB at 1550 nm wavelength. Its bandwidth in one band gap is about 81 nm.