Multicore Photonic Crystal Fibres Designing for Multiplexers and Demultiplexers Applications

Coupling properties are significantly affective when the three cores of a photonic crystal fiber are identical or non-identical, we analyze this numerically using COMSOL Multiphysics software 5.5 to gain insight into the nature of the coupling between these cores. A non-identical core produced by a slight mismatch in the propagation constants which are sufficient to suppress coupling between the cores or may strengthen the coupling between the cores because they sometimes have equal propagation constants. By manipulating the central core diameter (increase or decrease) compared to the diameter of the other cores can limit the strength of the coupling between the cores, the coupling efficiency and the coupling length of the cores become decreasing. Introducing anisotropy in three core diameters resulting in the modes of these cores are isolated in a certain region of the array of photonic crystal fiber, as a result, the coupling is suppressed between the modes of cores at the wavelength 1064nm while some coupling appears between modes of the cores in the wavelength 1550nm where the coupling coefficient is evaluated as a function of wavelength. The coupling between three cores reveals novel characteristics more than two cores and this useful for Multiplexers and Demultiplexers applications


INTRODUCTION
: Geometry structure of three-core photonic crystal fiber with three-ring hexagonal lattice: (a) insert core material; (b) insert the cladding material, (c) mesh free-triangular.
After knowing the effective refractive indexes of the modes, we can directly calculate the coupling length (L c ), it defines as a single power is exchanged between three cores, resulting in the weak overlap of the adjacent electric field of three cores. Therefore, a part of the light is confined into one core, then transfer to the other cores after propagation distance known coupling length can be calculated in equation (1)  represent the difference in propagation constants of even and odd modes, the refractive indexes that correspond to it are ( and ), as well as the wavelength λ that is used in the design. The coupling coefficient describes the percentage of power transferred between the three cores of the PCF, and relates to coupling length given by the following relation: . The numerical results showed that the coupling length L c for the identical core of 3.5μm and the non-identical cores of 3.2μm and 3.8μm, the effective mode indexes along the xpolarized field are for the even and odd modes, the differences in the effective refractive index for even and odd modes and the coupling coefficients at the λ=1550nm are shown in (Table 1). Fig. (2) represents the relationship between the coupling lengths, the coupling coefficients, and the core diameter for three structures with different designs that may be identical and non-identical in its cores, and this shown in (a) and (b) according to the results shown in (Table 1). These results indicate the possibility of designing a useful system for coupling or splitter with very short coupling lengths in (μm) useful in optical communications.

Result of identical and non-identical three cores PCF coupler
The rate of power transfer between the three cores PCF is known as the coupling length or beat length; consequently, the coupling length and the coupling coefficient are employed to determine the strength of that coupling that depends on the interactions between cores (Reichenbach et al., 2005;2007).
Supermode is a linear combination of the LP modes of each core, and the total number of nondegenerations supermodes is equivalent to the number N of cores in the system (Szostklewicz et al., 2016;Tan et al., 2015;Reichenbach et al., 2005;2007), where the mode of multicore fibers with the different propagation constants contains 2N modes, and the factor of two represents two polarization states of modes that referred to as the fundamental modes. The distribution of energy within these individual cores for each of the modes is nearly Gaussian with azimuthal symmetry. The mode solutions are of both even and odd modes for each polarization where the modes will propagate at little different speeds along the PCFs. A linear combination of the LP supermodes of each core with specific phase relation to the electric field distribution of three cores PCF is similar to the single-core fiber where the 1st supermode has considered the fundamental mode LP 01 mode that has the largest effective index is considered as lower-order mode while the supermode LP 11 and LP 02 are considered as a higher-order mode ( Tan et al., 2015;Kishi et al., 1989;Yu et al., 2006).
Three single power distributions of the six fundamental modes with polarized modes , (show the coupling between the central core and the surrounding cores), (the strong coupling between the surrounding cores than the central core), (suppress the coupling between the surrounding cores) for identical cores with diameters (3.5μm), display in Fig. 3c and non-identical cores with diameters of 3.2μm and 3.8μm with outer core diameters are (3.5μm), are displayed in Fig. 4c and 5c, when the light is initially interred to the central core. In mode, all cores have the same power and intensity, leading to all excited modes within the central core area. In mode, all the surrounding cores have propagation constants are nearly equal to the propagation constant of the central core. In mode, the surrounding cores are excited relative to the central core and the strong coupling between them because they have the same propagation constants nearly, therefore, power is limited in the outer cores. Fig. (3 a & b) show the effective mode index and Surface for Electric field distributions (V/m) of the x component for identical three cores PCF coupler for even and odd modes for a core diameter of 3.5 µm. From the numerical results in Fig. (3c) that represent the field distributions for identical line three-core PCF coupler design are LP 02 , LP 02 and LP 11 with the refractive indices are 1.4308, 1.4296 and 1.4282. The field distributions of identical three cores PCF shows supermodes with the field distributions of LP 02 and LP 11 that have a nonzero and zero fields in the center core respectively.

(a) (b)
First, concerning LP 02 modes, all the outer cores have propagation constants which are converted and almost equal to the propagation constant of the central core, where the mode field is more confined to the centre core. Therefore, the field distributions of LP 02 have lower propagation constants owing to the presence of the field in the center core, which means that the power is limited in the central core as shown in Fig. (3 c). Second, the fields distribution of LP 11 that have zero fields in the center core, when outer cores (higher-order modes) are excited relative to the central core and coupling between the surrounding cores become strong because of the propagation constants of these cores are equal with each other where the difference between the propagation constants of these cores is zero. The coupling strength normally depended on the mode; this means, the power is limited in the outer core. While Fig. (3d) represents the evolution of the power flow for the fundamental modes of the three cores shown in (c), where each core supports two propagation modes, and the initial input into the central core is at 1.55 µm. In this Figure, we show six lines of colors, the yellow line represents one of the central core modes, while the pink line represents the other, and the four rest of the colors represent the modes of the cores adjacent to the central core that is represented in light blue, red, green and blue. We notice one mode of the central core (yellow color) is coupled with two modes of the two adjacent cores (green and blue), or the two adjacent cores are strongly coupled together with their modes (light blue and red) without coupled to the mode of the central core or we may find the other mode of the central core (pink) is a weak coupling with the two adjacent cores Fig. (4) shown the effective mode index and Surface for Electric field distributions (V/m) of the x component for non-identical three cores PCF coupler for even and odd modes as in (a and b) that is represented with a core diameter of 3.2 µm. Fig. (4c) represents the field distributions for both cases of non-identical three cores PCF are LP 01 , LP 02 and LP 11 with the refractive indices are 1.4299, 1.4295 and 1.4265. Firstly, concerning LP 01 modes, all three cores have the same power where the intensity is equal among all cores and leads to all excited modes within the central core area. Secondly, concerning LP 02 modes, all the outer cores have propagation constants which are converted and almost equal to the propagation constant of the central core, where the mode field is more confined to the center core. Therefore, the field distributions of LP 01 and LP 02 have lower propagation constants owing to the presence of the field in the center core, which means that the power is limited in the central core. Thirdly, the fields distribution of LP 11 that have zero fields in the center core, when outer cores (higher-order modes) are excited relative to the central core and coupling between the surrounding cores become strong because of the propagation constants of theses cores are equal with each other where the difference between the propagation constants of these cores is zero and the coupling strength is morally depended on the mode, this means, the power is limited in the outer core. Fig. (4 d) expresses the evolution of the power flow for the three cores fundamental modes that are presented in (c), where each core supports two propagation modes when initially input into the central core is at 1.55 µm. We showed six color lines; the yellow line represents one of the central core modes coupled with two modes of adjacent cores (light blue and red), while the pink line represents the other mode of the central core, which not coupled to modes of adjacent cores. The other modes of the adjacent cores (green and blue) are strongly coupled together without coupled to the central core. Fig. (5) shows the effective mode index and Surface for Electric field distributions (V/m) of the x component for non-identical three cores PCF coupler for even and odd modes as in (a & b) that is represented with a core diameter of 3.8 µm. Fig. (5c) represent the field distributions for both cases of non-identical three cores PCF are LP 02 , LP 02 and LP 11 with the refractive indices are 1.4324, 1.4297 and 1.4288. Fig. (5d) represent the evolution of the power flow for the three cores fundamental modes that are presented in (c), the yellow and pink lines that represented the two modes of the central core either a strong or a weak coupling with two modes of adjacent cores as (blue and green) or the other modes of the adjacent cores (light blue and red ) are strongly coupled together without coupled to the central core. To evaluate the effect of anisotropy in all core diameters on the coupling properties, we designed PCF coupler consisting of variation in all core diameters, such as 3.5μm, 3.2μm, and (a) 3.8μm, with the same as the previous structural parameters at the wavelengths 1064nm to analyze the extent of its coupling properties. The numerical results in Fig. (6 a) showed that the unique three power distributions with the refractive indices are 1.4364, 1.4382 and 1.4397 and the cores are decoupled with their neighbors and remain isolated in their location at the wavelength 1.064nm. As a result, the difference between core diameters will generate a small mismatch between the modes of these cores and prevent the coupling between them, i.e. the mode is isolated in a certain region in the array of the PCF. Fig. (6 b) represents an evaluation of Fig. 6(a), where each core has two modes, i.e., six modes of three cores become, we observe the two modes of the central core that is representing by both yellow and pink lines, and the two modes of the adjacent core to the central core represented by the blue and green colors, as well as the other adjacent core to the central core also has two modes light blue and red. All three patterns of hearts we see through the shape are separate and completely, not coupled with each other.
By increasing the wavelength to 1550nm as shown in Fig. (6 c and d), we found through our numerical results the appearance of some coupling between cores of the three power distributions, i.e. the energy distributions with the refractive indices are 1.4268, 1.4293 and 1.4322 of the three cores are not unique and are some coupling between them as shown in Fig. (6 c). Fig. (6 d) represents an evaluation of Fig. (6 c), where also, as we mentioned earlier, each of the three cores contains two modes, i.e., we get six modes, we notice that two modes of the central core represented by the yellow and pink lines with a weak coupling of two modes of the one of adjacent cores that are represented by the light blue and red colors, or we may see the two modes of the other adjacent core of the central core which are represented by the colors of the blue and green lines, it propagates separately from the modes of the other cores and is not completely coupling with them, or perhaps we may find a weak association between the patterns of each of the three cores. From our result we conclude that the coupling efficiency is a function of the wavelength, where the coupling efficiency increases as the wavelength increased; this study is useful in multiplexers and de-multiplexers applications.

CONCLUSION
We performed numerical simulations using COMSOL Multiphysics 5.5, for three-core PCFs with either identical or non-identical cores to show how the coupling properties between cores is affected when we induce a small change in the diameter of the central core relative to the outer cores. The coupling length, the coupling efficiency and the power transmission between cores are calculated. The results showed that the coupling exists between the identical cores. While the coupling is suppressed or strengthened between non-identical cores. We introduced anisotropy in all core diameters to introduce small mismatches between neighboring cores, the results found that some coupling occurs between the cores at a wavelength of 1550nm and suppressed at the wavelength of 1064nm, this study is useful for multiplexing or demultiplexing applications.
This result can be therefore useful for space-division-multiplexed (SDM) optical transmission that is operating either without any coupling between modes or when the coupling exists between these modes; this means zero or small differential group delay (DGD). In the case of no coupling between modes, the modes can then be propagated independently and become separate modes while by presenting coupling between modes of the core, then mixed and can be separated by using multiple-input-multiple-output (MIMO) based digital signal processing (DSP) technique, this evidence super modes directly used as multiplexing or demultiplexing.