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Evaluated the coupling properties for seven cores photonic crystal fiber

Miami Mohammed1, Cornlie Denz1

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1 The university of Münster, Institute of applied physics, Münster, Germany

Abstract

Designed numerical simulation for seven cores photonic crystal fiber to evaluate the coupling properties, coupled power and field distributions of polarization modes between these cores by using Comsol multiphysics and coupled mode theory, this study includes the change many parameters such as amount of variation in core diameter, the separation distance, and these parameters play an important role to determine the inter-coupling that is related to crosstalk where coupling efficiency and then coupling length between seven cores are essential based on it. As well as the change in all diameter of the seven cores PCF leads to nonuniformity and suppress the crosstalk result from no coupling between modes and become as localized mode is limited in certain place from design, observe the effect of wavelength when there is nonuniformity in the central core diameter and this is achieved either small/ zero different group delay result from coupled or no coupled between cores resulting from nonuniformity and may become suitable for applications such as Multiplex or DE-multiplex. .
Keyword: multicore photonic crystal fiber, supermodes, coupled-mode theory, crosstalk, the separation distance, nonuniformity

Introduction

For many years, multicore photonic crystal fiber (MCPCF) has been widely used in many fields and has had an effective impact on the optical communication systems, fiber endoscopes, microwave photonics, fiber lasers, fiber sensors, coupling, switching and multiplexing 1, 2 owing to its higher information capacity compare with the single core. In (MCPCF) the cores either are sufficient individual to avoid crosstalk between them 3 or maybe appear strong coupling between the individual cores through evanescent field coupling of these cores result from the creation the supermodes among the cores has been proposed to increase the fiber capacity 1, 3, 4, supermodes can be defined as linear combinations of the LP modes guided in all individual core with orthogonal polarizers 4 and also, the supermodes can be used directly as the mode-division multiplexing(MDM) where supports different modes transfer with different differential group delays (DGDs) can be applied to achieve space-division-multiplexed (SDM )5 as new method to overcome on the capacity that is limit in the single mode fiber since modes coupling with each other over long distance fiber transmission and can be unraveled crosstalk between modes by using multiple-input-multiple-output (MIMO) receiver based digital signal processing (DSP) technique by using multicores as individual cores or groups of strongly coupled cores, then few cores can be support a reasonable number of modes and it possible to achieve low DGDs more simply. For weak coupling where all cores are identical therefore no mismatch occurs between the cores and achieved zero-DGDs while non-identical diameters by introducing a small nonuniformity in core diameters resulting zero or a small mismatch are enough to reduce the crosstalk between cores and consequently becomes decouple then a small different DGDS5-8. Over years, the studies interested about analysis of the supermodes in multicore PCF by using either numerical simulations or analysis method or numerical simulations and experimental, the analysis method can be used to analyze behaviors and properties of the supermodes with identical hexagonally distributed cores 1, 9-10, identical with circularly distributed cores 3, PCF with identical and non-identical cores coupled 11-13, non-identical core coupled only 14, numerical simulations for nonlinear dynamics all-optical switching PCF for non-identical core coupled with planer or triangular arrangement 2,15 numerical simulations and experimental for identical cores16, nonlinear dynamics all-optical switching with planer arrangement 17, 18. Also, can be used PCF as a multiplex de-multiplex application as 4-8, 19-21.

In this paper, focus on the analysis of the supermodes in seven cores PCF with hexagonally distributed cores by using numerical simulation COMSOL multiphysics software that is based on finite element method (FEM) as well as evaluated by using the coupled mode theory (CMT) and matrix operation, the analytical for both the propagation constants and modal distribution of supermodes are derived. We design numerical simulation of seven identical, non-identical cores PCF to discover an important role for crosstalk between adjacent cores to limit the coupling between seven cores PCF where introduced slight change in central core diameter towards increase or decrease relative to the outer cores diameters, also, the study includes introduce changing in separation distance between seven cores PCF towards incresing to know how to effect on the crosstalk between cores and finally, the change includes all diameters of seven cores PCF to reveal the nonuniformity in the fiber geometry and all these changes to know how to effect the coupling properties between cores PCF.

Theory

Approximate solutions for multicore fibers can finding analytical by coupled mode theory (CMT) when the structure extended to several cores neighbor to each other more than two cores and become complicate where the approximated result from the linear combination of the normal mode fields of the individual cores and by using the coupled-mode approximation as the linear coupling where all cores are identical, the coupled equation can be described for 2N+1 waveguides and assume only the nearest-neighbor coupling causing to the weakly overlapping modes then evaluates the field for 7coure PCF as 8, 12, 21.

idU_0/dz+?_0 U_0+k?_(n=1)^6?U_n =0 (1)

i?d U?_n/dz+?_0 U_n+kU_0+k(U_(n+1)+U_(n-1) )=0 (2)

Where U_n is represented as the amplitude of the mode field at the N cores, ?_0 is considered as the linear propagation constant of the single core in the absence of other cores, the coupling coefficient k is constant for all cores in waveguide arrays and N is the number of cores in waveguide arrays that is running from ?N to N. The solution of the linear coupling waveguide array can be solved analytically according to the initial condition when only one waveguide is primarily excited as U_0(z=0) =? u?_0, U_n(z=0) =0 when n ?0, then the solution becomes as 22-24

U_n (z)=u_0 (?i)?^n J_n (2kz) exp?(i?_° z) (3)

Where J_n is represented the Bessel function of the order N. Physically, when the optical wave propagates along the waveguide, the power become spreads among neighboring cores of waveguide with a symmetric modes, where the intensity distribution is as J_n^2 (2kz) at any propagation distance z, show at the distance like 2kz=2.405, Bessel function J_0 is vanished then all the power becomes disappeared from the origin waveguide as ?_(n=1)^6?u_n =0 with the assumption the field is zero in central core or origin waveguide u_0=0 then become the solution only to outer cores where the field distribution for the LP_11 supermode as below 21, 22

i ?dU?_n/dz+ u_(n+1)+u_(n-1)=0 (4)

Also, at J_0 a portion of the power will reappear with additional propagation, this solution acts as the inter-waveguide diffraction or discrete diffraction where is a straight result of the diffraction that is occurred inside waveguide arrays can still spread the optical beam from one core to the other in waveguide arrarys with low power, as a result, the coupling between cores of waveguide arrays 22.

Since supermode as a linear combination of the LP modes of each individual cores and the total number of non-degeneration supermodes is equivalent to the number N of cores in the system 22, 25-26. Where the mode of multicore fibers with the different propagation constants contains 2N modes and the factor of two is represented two polarization states of modes that are referred to the fundamental modes so the distribution of energy within these individual cores for each of the modes is nearly Gaussian with azimuthal symmetry. The mode solutions are of even and odd mode for each polarization where the modes will propagate at little different speeds along the PCFs then results from the splitting in the mode effective indices and the power in both core is beating along the length of the fiber. The 1st supermode has considered the fundamental mode is the LP_01 mode that has the largest effective index, while the second and third supermodes are the LP_11 mode, the fourth and fifth supermodes are the LP_21 mode, the sixth supermode is LP_02 mode and the seventh mode is LP_31 mode 1, 11, 27.

While in case the nonlinear coupling between waveguide arrays can be described as following down and by considering the nonlinear effect for the one waveguide in the arrays to the nearest neighbor is as ignored, this equation is defined as discrete nonlinear Schrödinger Equation 22, 24,28.
i (dU_n)/dz+ ?_0 U_n+k(U_(n+1)+U_(n-1) )+?|U_n |^2 U_n=0 (5)

Where ? represented the nonlinear parameter as (?_° n_2)?(cA_eff ), ?_° consider the frequency that is associated with the waveguides modes, n_2 is the Kerr parameter, c is light speed in the vacuum and A_eff is the effective area of these modes, the equation (5) is defined as discrete nonlinear Schrödinger equation. The solution of the nonlinear waveguide arrays, when increasing the input intensity becomes as uncoupled power then the Kerr nonlinear effect began to effect on the propagation of the waveguide arrays and could not solve analytically then field distribution within waveguide array for high intensity exceeding the power threshold, the solution becomes as

a_n (z)=A_0 sech (X_n/X_0) exp?(i?_z+?2k?_z) (6)

X_n=nd has represented the position of n_th in waveguide arrays, X_0 has represented the width of the discrete spatial soliton and the latter is relatively big, so the spatial soliton tails are spread over a large number of waveguide arrays. The spatial soliton is a product of the balance between the discrete diffraction and nonlinear self–focusing effect, where the power is constant along total propagation distance 18, 24-29, 31. Also, the strong coupling can occur between three cores of waveguide arrays instead of several waveguide arrays as the discrete spatial soliton this means between the central cores and one or two nearby waveguides 22, 29, 30. Further, when increasing the input power, the soliton becomes as nonlinear modes localized over only a few neighboring modes or a single mode and competently decoupled from the residual waveguide arrays, therefore discrete spatial soliton can be formed from certain input parameters but with complex phase structure 22, 30.

When the waveguide or cores arrays is in 2D dimensions and by supposing only the nearest-neighbor coupling then calculates the mode amplitude for N waveguide arrays for the equation (5) can be described coupled-mode equation in a matrix form as

d/dz a (z) = iMa (7)

Where a (z) = ? a_0 (z),a_1 (z),…..a_(n-1) (z)?^T is the mode amplitude of the field at N core, a= ? a_0 ,a_1 ,…..a_(n-1) ?^T is represented the complex amplitude of the electrical field of the nth core, M is represented the coupling matrix (2N+1) × (2N+1) with the propagation constant ? ?=?+?|a_n |^2 of the single mode and k coupling coefficients between adjacent cores. Also, the matrix based on of structure distribution of waveguide arrays as linear or triangle 1, 15, 21, 25, 27, 32-33.

or

The coupling between waveguide arrays have shown demonstrated a host of a novel physical phenomenon as discrete diffraction and discrete spatial solitons effects depended on both of the discrete nature of the coupling between waveguide arrays and the refraction index of the medium. Discrete diffraction occurred when are evanescently coupled between identical waveguide arrays at a low power, the couple from the central core waveguide to the neighboring cores with refraction indices are constant. But the situation is different when are coupled between the non-identical waveguide arrays where significant nonlinear index changes depended on the intensity and decoupled power from the central core to the neighbor’s cores result as known as discrete self–focusing18. Also, by increasing optical power due to the self-localization of light and then elimination of diffractive broadening of pulse, when the input beam is enough strong result balance between nonlinearity and diffraction and light propagate becomes localized soliton in the limited region of the waveguide arrays where the beam is preserving its shape during propagation as discrete spatial soliton 22, 24-29.

Usually, seven cores become non-identical PCF either varies in indexes refractive core and cladding or may be in cores diameters or in fiber dimensions 7, in this study we assume to vary in diameters the central cores relative to the diameters the outer cores which occurring seized constant then introduce mismatch between cores, in all cases we supposing the cores are single-mode and only supports the fundamental mode LP_01, the cores, the propagation constant is ?_°, the separation distance between cores them is D and the coupling coefficient k between the N cores array are calculated using approximation and identical cores(as weakly guiding) 5-6, 21, 32, 34.

K=?2? U^2/(RV^3 ) (K_° (WD/R))/(K_1^2 (W)) (8)
R is represented the radius of the, ?= n_co^2-n_clad^2/?2n?_co^2, n_co and n_clad are the refractive index of the core and the cladding and ? is represented as the refractive index difference between core and cladding, V=k_° Rn_co ?(2?)?^(1/2) is represented the eigenvalue problem that is described the fundamental mode of the core involves the parameters U and W can be found by solving equation below

UJ_1 (U)/J_° (U)=WK_1 (W)/K_° (W) (9)

U= R?(k_°^2 n_co^2-?_2)?^(1/2), W= R?(?_2-k_°^2 n_cl^2)?^(1/2) (10)

Where k_°=2?/? is the wave number, K_°, and K_1 are represented the zero and first-order modified Bessel functions of the second kind J_°, and J_1 are the zero and first order Bessel functions of the first kind respectively 12, 16 , Then the propagation constant ? possible to determine from the eigenvalue problem by assuming that U^2+W^2=V^27, 21, 34. Also, when all the cores are identical there is no mismatch between them and have the same propagation constant ?_°, therefore zero-DGD can be achieved when dk/d?=0, then the eigenvalues of the multicore PCF is as

dk_n/d?=dk_1/d?=dk_2/d?,…….. (11)

The equation (11) confirms that the group velocities are equal of all N supermodes when all derivatives are estimated at the central frequency ?=?_°6, 8. The situation is different when the cores are non-identical core where the coupling coefficients are different as k_0 ?k_n as k_0n=(k_°^2)/2? ?_(-?)^??(n_co^2-n_clad^2 ) F_1^* (x,y)F_2 (x,y)dxdy Where n_co has represented the refractive index of the core, n_clad the refractive index of the cladding, Fi(x,y) is represented denoting the radial distribution of the mode (i = 1, 2) due to mismatch between propagation constants of these core6, 35, then the coupling coefficient is as 6, 8.
k_0n=(??(2??_0)?^(1/2) ? U?_n U_0)/(R_n V_n )×(k_° ((W_n D_0n)/R_n ))/(K_1 ?(W?_n ?)K?_1 (W_0 ) )×{(W ?_n ) ?K_° (W_0)I_1 W ?_n+W_0 K_(1() W_(0)) I_(°() (W_(n)) ) ?/(W_n^2 ) ?U_0^2} (12)
Where (W_n ) ?=W_n R_0/R_n, K_0, and I_0 are represented the modified Bessel functions of the first and second kind, R_n are represented the core radii, the normalized index difference is ?_0=(n_(co,0)-n_(clad,0))/n_(co,0), the normalized frequency for the core n is V_n=k_° R_n n_(co,n) ?(?2??_n)?^(1/2) then V^2=W^2+U^2 therefore W_n=R_n ?(?_n^2-k_°^2 n_(clad,n)^2)?^(1/2) and U_n=R_n ?(k_°^2 n_(co,n)^2-?_n^2)?^(1/2), n_(co,0) and n_(clad,0) are represented the core and cladding refractive indices, k_° is the free-space wavenumber and D_0n is the distance between cores 0, n, Then the propagation constant ?_n possible to determine from the eigenvalue problem by assuming is as follow below

?_n=??_n+? ? (13)

Where ? ?=k_° (n_co (?)+n_clad (?))/2 is dropped out from derivative the equation relative to angular frequency than the equation (13) only depended on ??_n involved a smaller different between the propagation constants and is related to the average of all the group velocities that are associated with multicore PCF 6, 8. Also, non-identical cores multicore PCF can be evaluated to zero or small differential group delay DGD between supermodes even though exist some coupling between them and according to the equation (14) DGD between i^th and j^th super-modes can be written as6.

?DGD(?_(i,j))=(d?_i)/d?-(d?_j)/d?=?_(i=1)^2???(a?_n^i-a_n^j).dk_n/d?? (14)
Where the parameters a_n^i, a_n^j are related to the supermode propagation constants ? both of the i^th and j^th supermodes and dk_n/d? represented the coupling coefficients for k_n (n=1,2,…..).

3. Simulation and Discussion

By using Comsol Multiphysics software, we designed numerical simulation that is based on finite element method, the designer includes identical and non-identical seven-core PCF that is consisted of five rings of air holes are arranged in a hexagonal lattice in the cladding with hole diameter d_(hole )=4.48 um, pitch ?=5.6um and the air-filling fraction is d ??=0.8 with different diameters of the central core d_(core )=3.5,3.2 and 3.8 um, while the outer cores is constant d_(outer core )=3.5um in all cases, the separation distance between adjacent cores is D=2um and the difference between the central core and the outer cores is ?d=0,±0.3 for different wavelengths 1.064 and 1.55um for silica material showsTabel A.1in appendix.

3.1 Design identical Seven-Core Photonic Crystal Fiber

3.1.1 Identical core

For identical core d_core=3.5um,? d?_outer=3.5um and ?d=0 um at the wavelength 1.064um, the different effective index between even and odd mode is 0.0005 and has coupling length is L_c=?/2(n_eff^e-n_eff^o) = 1.064um this is shown in Fig. 1-3.

Fig.1: The geometry structure of the seven core photonic crystal fiber

Fig. 2: Electric field in (a, c) and intensity profile in (b, d), x component (V/m) even and odd modes PCF coupler

Fig. 3: Power flow of the fourteen fundamental modes of identical seven cores PCF coupler when the input is in the central core fiber

For identical core d_core=3.5um, d_outer=3.5um and ?d=0 um at the wavelength 1.55um, the different effective index between the even and the odd mode is 0.0017 and has coupling length is L_c= 455.88um, this is shown in Fig. 4-5.

Fig. 4: Electric field in (a, c) and intensity profile in (b, d), x component (V/m) the even and the odd modes PCF coupler

Fig. 5: Power flow of the fourteen fundamental modes of identical seven cores PCF coupler when the input is at the central core

3.2 Study the coupling nature between identical seven cores PCFs
To understand the nature of the coupling between seven cores of PCFs by calculating the power amount or energy distribution in all cores after propagation distance along the fiber length. When all the diameters of cores are identical, the power will periodical oscillates between cores as a function of the propagation distance z. It possible to describe the percentage of power transferred between the seven cores as the coupling efficiency that is connected to crosstalk, and the rate of power transfer between the seven cores as the coupling length or beat length, consequently the coupling length and the coupling efficiency are employed to determine the strength of coupling that is depended on the interactions between cores.

The coupling between seven cores PCFs is different from the single core because the central core is surrounded with six adjacent cores and presence further cores due to increasing the rate at which power is transferred of these cores, shows from Fig. 6-7 for identical seven cores PCFs, the power in all cores as function the propagation distance z and more the power is focused in the central core where input initial power while the power in the outer six cores is labeled where nearly 60% of the coupled power will oscillate back and forth between the center core and the adjust six cores, while around 40% from the residual power will be retained in the central core where initial input it. This behavior is similar to the case the coupling between non-identical two cores PCFs, but power amount which retains to the initial input core is larger about 90% while 10% coupled power between cores. Therefore, it shows the amount of coupled power will be decreased by the interactions with adjusting six cores, and coupling length will decrease when presence adjacent cores from coupling length of two cores PCFs.

Also, shows from Fig. 6- 7 the field distributions for weakly guiding and identical seven cores PCF, since these supermodes can be described as a linear combination of the LP modes of each individual cores with specific phase relationship with the electric field distribution of seven cores PCF are similar to the single core fiber where the first mode is the fundamental mode LP_01 that is degenerated HE_11 modes which have the largest effective index) are consider as lower-order mode, while the second and third modes can be described to LP_11 mode that includes degenerate as TE_01, TM_01, and HE_21 modes) are considered as higher- order modes, The fourth and fifth modes is LP_21 includes degenerate to HE_11 and HE_31 modes, while the sixth mod is LP_02 includes degenerate to HE_12 modes and the seven mode is LP_31 includes degenerate to HE_31, the coupling stronger between higher orders modes is more than lower-order modes (fundamental mode), the simulation results in identical cores case is similar to 11-12, 21, 37 as numerical but using different the wavelengths add to this work to show the effect on the coupling properties and with different design.

Supermodes with the field distributions of LP_01 and LP_02 that have a non-zero field in the center core and while the residual supermodes are quite unaffected. For the LP_01 mode, all seven cores have the same power where the intensity is equal among all cores and leads to all excited modes within the central core area. For the LP_02 mode, all the outer cores have propagation constants are converged and nearly 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_01and LP_02 have lower propagation constants owing to the presence of the field in the center core, this means, the power is limited in the central core. While the fields distributions 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 and the coupling strength is morally depended on the mode, this means, the power is limited in the outer core. Otherwise, in each surrounding cores of central core become the mode coupling is strongest between the 3 or maybe 4 nearest neighbor cores this shows from results of simulation by Comsol multiphysics, all these results can be useful to tailor the near field of outer cores with the central cores leads to important increase in multicores power handling could be realized and simply scalable to the larger of waveguide arrays and multicore PCF.

Fig. 6: The power evolution of the fourteen fundamental modes of the identical seven core PCF fiber for linear polarization modes LP_01in (a) LP_02 in (b), LP_11 and nearest neighbor outer cores in(c-f), and from down, the power in each core of seven cores of when initial input is in the central core where the weak coupling with the central core at 1.064um

Fig. 7: The power evolution of the identical seven core PCF fiber for linear polarization modes LP_01in (a), LP_02 in (b) and nearest neighbor outer cores in (c-e) LP_11, the power in each core of seven cores of when initial input is in the central core where the weak coupling with the central core at 1.55um

3.3 Design non- identical Seven-Core Photonic Crystal Fiber

Non-identical core

For non-identical core d_core=3.2um, d_outercore=3.5um, ?d=-0.3 um is represented the difference between the diameter of the central core and the other core at 1.064um, the different effective index between the even and the odd mode is 0.0004 and has coupling length is L_c=1,330um, shows in Fig. 8-9.

Fig. 8: Electric field in (a, c) and intensity profile in (b, d), x component (V/m) even and odd modes PCF coupler

Fig. 9: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when the input is at the central core at1.064um

2- d_core=3.2um, d_outer=3.5um, ?d=-0.3um has represented the difference between the diameter of the central core and the other core at 1.55um, different effective index between the even and the odd mode is 0.0015 and has coupling length is L_c=516,66um, shows in Fig10-11.

Fig. 10: Electric field in (a, c) and intensity profile in (b, d), x component (V/m) even and odd modes PCF coupler

Fig. 11: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when the input is at the central core at 1.55um
3- d_core=3.8um, d_outercore=3.5um, ?d=+0.3 um is represented the difference between the diameter of the central core and the other core, the different effective index between the even and the odd mode is 0.0006 at 1.064um and has coupling length is L_c=886.6um, shows in Fig. 12-13

Fig. 12: Electric field in (a, c) and intensity profile in (b, d), x component (V/m) even and odd modes PCF coupler

Fig. 13: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when the input is at the central core at 1.064um
4- d_core=3.8um,? d?_outercore=3.5um, ?d=+0.3 um is represented the difference between the diameter of the central core and the other core at 1.55um, different effective index between the even and the odd mode is 0.0018 and has coupling length is L_c=430.55um, shows in Fig14-15.

Fig. 14: Electric field in (a, c) and intensity profile in (b, d), x component (V/m) even and odd modes PCF coupler

Fig.15: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when the input is at the central core at 1.55um
3.4 Study the coupling nature between non-identical seven cores PCFs
For non-identical cores, the small change in central cores diameters will result from asymmetry in refractive index between cores and thus effect on the coupling properties of the seven cores PCFs, as well as the wavelength, and the separation distance between cores also, have a large effect on coupling properties. Consequently, this change between cores causes a decrease or increase in magnitude both the coupling efficiency and the coupling length of these cores from their value when these cores PCFs are identical. For the small coupling efficiency, then power oscillates between seven cores is small and the coupling length calculates stops on crosstalk or the strength of coupling between cores. The introduction changes in central core diameters cause nonuniformity in core size, then decrease the crosstalk in cores or another hand may be the coupling is strong. The Fig.16- 17 is represented the power evolution in non-identical seven cores, when one of adjusting six cores diameters has a diameter close to the central core this means have equally propagation constants, shows the coupling behavior of a seven cores PCFs is approximately similar to the two core PCFs in case of identical cores while when the two cores have diameters nearly to the central core where appear behave similar to the three cores PCFs when identical cores and this depending on the matrix orientation for coupling three cores either a line or a triangle while the residual cores appear behavior similar to the case non-identical two cores PCFs where contributing to the power exchange is very slight , the simulation results in non-identical cores case starts almost similar to 11 as numerical as well as using different the wavelengths add to this work to show the effect on the coupling properties and with different design.

Fig. 16: The power evolution of non-identical seven cores PCFs ±0.3, in case the strong coupling when one of the six adjust cores have the diameter closer to the central core in LP_21 (a) or two of the six adjust cores have the diameter closer to the central core depending on the orientation of the three matrix coupling cores either linear or triangle in LP_31 (b and c), while the other case without any coupling with the central core only the coupling is with the nearest neighbors in the array in LP_11 (d and e) at the wavelength 1.064um.

Fig. 17: The power evolution of non-identical seven cores PCFs ±0.3, in case the strong coupling when one of the six adjusted cores have the diameter closer to the central core in LP_21 (a) or two of the six adjust cores have the diameter closer to the central core depending on the orientation of the three matrix coupling cores either linear or triangle in LP_31 (b) , while the other case without any coupling with the central core only the coupling is with the nearest neighbors in the array in LP_11(c and d) at the wavelength 1.55um.

On other hands, shows when increasing in the core size will find the mode coupling is stronger since the higher order modes that are supported the larger cores and also the lower order mode becomes strong overlap, as a result, the reduction in the separating distance between the mode fields of the adjacent cores. While decreasing in core size will reduce the number of modes then the mode coupling becomes weaker causes and increased in the spacing distance between the mode fields of the adjacent cores. Therefore, the small variation in core size increases or decrease could be caused changed in crosstalk since the number of guided modes is changed. Also, the influences of the strength inter-core coupling of the guided modes become stronger. The results from numerical simulation when is a smaller change in core size can suitably limit crosstalk to realize robust transmission 11-12.

Also, the relation between the coupling length and the difference between the diameter of the central core and the outers cores is ?d, when increasing the difference between the diameter of the central core and the outer cores, shows both of coupling length and the coupling efficiency will change depending on the degree of change in the core size and also on the wavelength. So, coupling length is decreased more dramatically at shorter wavelengths than greater wavelengths as shown in Fig. 18.

Fig. 18: The variation between the coupling length and the difference between the central core and the outer cores of seven cores PCF

Adding, the relation between the coupling length and wavelength is inverted, of seven cores PCF shows in Fig. 19. By increasing the wavelength, the coupling length decreases, the coupling length at the wavelength 1.064 is greater than 1.55um. On other hands notice from this Figure, the coupling length when the cores are non-identical is greater or smaller than identical cores according to the change in central core diameter.

Fig. 19: The relation between coupling length and wavelength for identical and non-identical seven cores PCF coupler

In addition to that, the coupling characteristics of seven cores PCF couplers of the wavelengths 1.064 and 1.55um show in Fig. 20, the coupling length of the wavelengths 1.55um is lower than1.064 um. Hence, it is potential to design significantly shorter MUX-DEMUX PCFs of seven cores PCF using coupling lengths of a few millimeters and compared this to standard optical fiber couplers that use coupling lengths of more than a few tens or even hundreds of millimeters this result similar to38 with different design and different wavelengths.

Fig. 20: Coupling characteristics of seven cores PCF couplers at the wavelengths 1.064 and 1.55um

4. Effect the separation distance on the coupling between seven cores PCF coupler

The irregularity of the separation distance between the cores will effect on the coupling length and then on the coupling efficiency when the core diameters are identical and non-identical cores. For identical core, the diameter of the cores are constant, increasing the separation distance between the cores due to large separation between mode Fields then reduced coupling between the neighboring fibers, and may be modes becomes localized in the restricted region of the multicore PCF. While for non-identical cores, the changing in the separation distance between the cores become more or less sensitive to core diameter mismatch depending on these cores either closer or farther from each other when the separation distance is lower or larger. So increasing in separation distance has opposite result from decreasing, therefore, the net effect is minor. A non-identical core cause’s mismatch where inducing a small variation in a core diameter will decrease the inter-core mode coupling (crosstalk) between neighboring cores significantly, but also at the same time perhaps strong inter-core coupling can be introduced when propagation constants of some modes of these cores with different diameters are almost identical and nearly phase mismatch is zero between these modes. Therefore, a small variation in core diameter and increase in separation distance can appropriately limited crosstalk between cores and will probable permits to achieve a strong transmission 12 this reference try to point out what happens w changes when the separation distance changes to the identical cores, while we tired here to change the separation distance for identical and non-identical cores as numerical as well as using different the wavelengths to see the effect of the separations on the coupling properties in order to have a clear vision of how to change these properties.

From numerical results of the numerical simulation, Fig 21 shows how to change the coupling between cores with change the separation distance for identical and non-identical seven cores photonic crystal fiber at the wavelengths 1.064um and 1.55um.
Firstly, all identical seven cores d_core=3.5, d_outer=3.5um, n_core=1.445 , n_cldd=1.44, hole diameter , d_hole =4.48, ?,= 5.6um, the separation distance D=1.5, 2 and 2.5um, the different effective index between even and odd mode are 0.0002, 0.0005 and 0.0006 and the coupling length are L_c=2.660 um, 1.064um and 866.66um at the wavelength 1.064um . While for the wavelength 1.55um are the different effective index between the even and the odd mode are 0.0003, 0.0017 and 0.0018 and the coupling length is L_c=516.66 um, 455.88um and 430.55um, Fig. 21a-f Shows from theoretical results of simulation numerical, when all the identical cores and change in the separation distance towards increases, shows the coupling length between the modes is decreased, subsequence the coupling length at the wavelength 1.064um is larger than the wavelength 1.55um.

Secondly, for non-identical cores 3.5-0.3, at the wavelength 1.064um, the separation distance D=1.5, 2 and 2.5um, the different effective index between even and odd mode are 0.0003,0.0004 and 0.0006 and the coupling length is L_c=1,773um, 1,330um and 886.66 um. While for the wavelength 1.55um are the different effective index between the even and the odd mode is 0.00014, 0.0015 and 0.0016 and the coupling length is 553.57,516.66, 484.37um, Fig. 21 g-l

Third, for non-identical cores 3.5+0.3 at the wavelength 1.064um, the different effective index between even and odd mode are 0.00055, 0.0006, 0.0009 and the coupling length is L_c=967.27um, 886.6um, 591.11um. While at the wavelength 1.55um, the different effective index between the even and the odd mode is 0.0016, 0.0018, 0.0019 and has coupling length is L_c=516.88um, 430.55um, 407.89um, Fig. 23 m-r. Also, the results from the numerical simulation for the non-identical cores are 3.5 ±0.3 at the wavelengths 1.064um and 1.55um, where the change in the separation. When increasing the separation distance, the coupling length is decreased. As well as, the coupling strength between the neighboring cores becomes more than identical cores, but coupling strength decrease with increasing separation distance and is lower at the wavelength 1.55um than 1.064 um.

4.1- For identical core (all cores are 3.5um) at the wavelength 1.064um.

1-D=1.5um, the different effective index between the even and the odd mode is 0.0002, L_c=2,660 um

Fig. 21a: Power flow of the fourteen fundamental modes of identical seven cores PCF coupler when a change in separation distance

2-D= 2um, the different effective index between the even and the odd mode is 0.0005, L_c=1,064 um.

Fig. 21b: Power flow of the fourteen fundamental modes of identical seven cores PCF coupler when a change in separation distance

3-D=2.5um, the different effective index between even and odd mode is 0.0006, L_c=
866.66um

Fig. 21c: Power flow of the fourteen fundamental modes of identical seven cores PCF coupler when a change in separation distance

4.2-For identical core (all cores are 3.5um) at the wavelength 1.55um

1-D=1.5um, different effective index between even and odd mode is 0.003, L_c= 516.66um

Fig. 21d: Power flow of the fourteen fundamental modes of seven cores PCF coupler when a change in separation distance

2-D= 2um, the different effective index between even and odd mode is 0.0017, L_c=455.88um

Fig. 21e: Power flow of the fourteen fundamental modes of identical seven cores PCF coupler when a change in separation distance

3-D= 2.5um, the different effective index between even and odd mode is 0.0018, L_c=430.55um

Fig. 21f: Power flow of the fourteen fundamental modes of identical seven cores PCF coupler when a change in separation distance

4.3- For non-identical core 3.5-0.3at the wavelength 1.064um

1-D=1.5um, the different effective index between the even and the odd mode is 0.0003, L_c=1.773um

Fig. 21g: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

2-D=2um, the different effective index between the even and the odd mode is 0.0004, L_c=1.330um

Fig. 21h: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

3-D=2.5um, the different effective index between the even and the odd mode is 0.0006, L_c=886.66um

Fig. 21i: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

4.4-For non-identical core 3.5-0.3at the wavelength 1.55um

1-D=1.5um, the different effective index between the even and the odd mode is 0.0014, L_c=553.57um

Fig.21j: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

2-D=2um, the different effective index between the even and the odd mode is 0.0015, L_c=516.66um

Fig. 21k: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

3-D=2.5um, the different effective index between the even and the odd mode is 0.0016, L_c=484.37um

Fig. 21l: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

4.5-Third, for non-identical core 3.5+0.3at the wavelength 1.064um

1-D=1.5um, the different effective index between the even and the odd mode is 0.0005, L_c=967.27um.

Fig. 21m: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

2-D=2um, the different effective index between the even and the odd mode is 0.0006, L_c=886.6um

Figure (6.21n): Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

3-D=2.5um, the different effective index between the even and the odd mode is 0.0009, L_c=591.11um

Fig21o: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

4.6-For non-identical core 3.5+0.3at the wavelength 1.55um

1-D=1.5um, the different effective index between the even and the odd mode is 0.0016, L_c=516.88 um

Fig. 21p: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

2-D=2um, the different effective index between the even and the odd mode is 0.0018, L_c=430.55um

Fig. 21q: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

3-D=2.5um, the different effective index between the even and the odd mode is 0.0019, L_c=407.89um

Fig. 21r: Power flow of the fourteen fundamental modes of non-identical seven cores PCF coupler when a change in separation distance

5. Study effect of nonuniformity in cores sizes on coupling properties

Effect of nonuniformity in cores sizes on coupling properties and crosstalk for seven cores PCF, We can be understood more clearly, the changing in field distribution of modes, as a result, the dramatic effects that are occurred from nonuniformity in core sizes and how is this effect on the coupling properties between seven cores PCF. When the all cores diameters for the PCF are different significantly, shows the modes for these cores will decouple into those of the individual cores and the presence of localized modes in a disordered seven cores PCF where note the energy distribution of these cores are unique, decoupled and localized mode is limited in the arrays region mode as discrete spatial soliton (localized soliton) 11.

Results from the numerical simulation for different wavelengths Fig.22 and 23 decoupled modes and intensity profiles for seven cores PCF as a result to fourteen non-degenerate modes for seven core with diameter 3.2, 3.5, 3.47, 3.48, 3.49, 3.51and 4.52 um showsTabel A.2in appendix. When the light incident on the single core of the PCF will occur couple only for just one mode and don’t occur coupled with others cores of the PCF, the field growth has one of the modes and then the energy distribution of mode becomes no longer the strong function of z. Otherwise, the difference between adjacent cores diameters will generate the mismatch between modes of these cores and subsequent this prevent the coupling between them. We do this by Comsol simulation for two the wavelengths are 1.064 and 1.55um and display for unique energy distributions of these modes seven cores PCF from fourteen non-degenerates modes of a decoupled seven cores PCF, also, shows the coupling efficiency is decreased when increases the changing in core diameter and with decreasing in the wavelength.

Fig. 22: Decoupled modes and Intensity profiles for seven cores PCF (a- g) and (h-n) with different of adjacent cores diameters at 1.064um.

Fig. 23: Decoupled modes and Intensity profiles for seven cores PCF (a- g) and (h-n) with different of adjacent cores diameters at 1.55um.

6. Conclusions
Design seven cores PCF by numerical simulation Comsol multiphysics to study the coupling strength between these cores when are identical or non-identical cores with the separation distance between cores is small and constant, the simulation results showed. Firstly, in the case identical cores the coupled between seven cores PCF is similar to non-identical two cores PCFs but power amount which retains to the initial input core is larger about 90% from seven cores about 40% coupled power between cores this means amount of coupled power will be decreased by presence adjacent cores and coupling length also decrease from coupling length of two cores PCFs and supermode appear with polarization modes LP_01, LP_02 and LP_11. Secondly, in the case non-identical cores the coupled between seven cores PCF where the small change in central cores diameter towards increase or decrease will result from asymmetry in refractive index between cores and thus effect on the coupling properties includes both of this change between cores causes a decrease or increase in magnitude both the coupling efficiency and the coupling length depended on crosstalk or the strength of coupling between cores leads to no coupling between the central cores and the outer cores(only the coupling between adjusted outer cores) or strong coupling when the diameter of one or two of the outer cores have the same propagation constant to the central core results coupling between them as two or three identical cores then appears new of the polarization modes as LP_21, LP_31,and LP_11. On another hand design, non-identical cores PCF may be no or strong coupling between cores, therefore, these modes where the zero or small different group delay. Thirdly, The irregularity of the separation distance between the cores will effect on the coupling efficiency then coupling length when the core diameters are identical and non-identical cores, showed from numerical results, the length coupling in case decreasing the diameter of the central core is larger than length coupling in case increasing the diameter of the central core, but the overall result length coupling decreases with increasing the separation distance for different wavelength. Therefore, a small variation in core diameter and increase in separation distance is limited crosstalk between cores and permits to achieve a strong transmission and finally, Moreover, design seven cores with different diameters from each other results nonuniformity in the core diameters due to suppress the crosstalk between seven cores, substantially the cores becomes no coupled, as well as, the mode can be propagated with zero group delay and become as localized mode, therefore, the nonuniformity in core diameters leads mismatch between neighboring cores where the interacting or coupling between cores depends on parameters such as the change magnitude in cores diameters and the wavelength. From the numerical results find at, the wavelength 1.064um the energy distributions of seven cores is quite unique for seven cores PCF, while at the wavelength 1.55um is shown energy distributions almost single for all seven cores but in this wavelength some coupling between the outer cores PCF not shown in the wavelength 1.064um. Generally, all these couplings can be provided to the characteristic could be needed for multiplexing or demultiplexing of these modes depended on the amount different group delay is zero/ small DGD.

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Appendix

Tabel A.1: The parameters of design three cores PCFs of silica glass material
Parameter Value
Wavelength 1.064 and 1.55um
The central core diameters for identical and non-identical cores
d_core=3.5um , 3.2 and 3.8

the outer cores diameter d_(outer cores)=3.5 um
The separation distance between cores D=2 um
n_co hole diameter d_hole =4.48 um
n_cl hole pitch ?= 5.6 um

Tabel A.2: The parameters of design three cores PCFs of silica glass material
Parameter Value
Wavelength 1.064 and 1.55um
The diameters of seven cores
3.2, 3.5, 3.47, 3.48, 3.49, 3.51and 4.52
The separation distance between cores D=2.5 um
n_co hole diameter d_hole =4.48 um
n_cl hole pitch ?= 5.6 um

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