Design and Modal Analysis of Photonic Crystal Fiber for Dispersion Compensation over Broadband Range

In this paper we have presented the four new investigations using different structures of Photonic Crystal Fiber (PCF) for broadband communication applications and narrows down on the design which can provide largest flat negative dispersion. The new structure model is optimized based on the combination of modal properties and dispersion compensation. The results were recorded over a transmission frequency range from 190.95 THz to 245.73 THz i.e. 1.22 μm to 1.57 μm with a dispersion -704.62 ps/nm/km. The results obtained from the new different PCF structures are compared with reported results for different 2D PCF models so far.


I. INTRODUCTION
Dispersion is a key factor for limiting the development in rapid transmission rate in communication field using optic fiber.For high speed and large bandwidth application in optic fiber communication, dispersion characteristic should be uniform or have minimum slope (flattened) throughout the wavelength [1].In the literature, design of different 2D PCF structures by compensating dispersion by changing geometric parameters of the structure and Refractive Index (RI) have been reported.However, conventional hexagonal PCF does not have air hole at the center of the core.In hexagonal PCF structures may have single air hole or array of air holes at the core.Feroza Begum et.al. [2] studied conventional hexagonal PCF structure for negative dispersion where D = -100 ps/nm/km at 1550 nm wavelength with high losses.Makoui et.al. [3] used generic algorithm to minimize pulse broadening over large wavelength.Varshney et.al.[4] used microstructure of PCF for residual dispersion compensation to achieve flattened negative dispersion.
They reported ultra-flattened negative dispersion of -98.3 ps/nm/km and ∆D= 1.1 ps/nm/km over S to L wavelength band i.e. wavelength bandwidth of 1.48 µm to 1.63 µm.Similarly, Franco et.al. [5] used microstructure of PCF and investigated ultra-flattened negative dispersion of -179 ps/nm/km and ∆D= 2.1 ps/nm/km over S to U wavelength band i.e. wavelength bandwidth of 1.48 to 1.675 µm.The design by Silva et.al.[6] demonstrates higher average dispersion -212 ps/nm/km and ∆D= 11.9 ps/nm/km over E to U wavelength band i.e. wavelength bandwidth of 1.35  inner (core region) and outer (Cladding region), there is a possibility to adjust effective refractive index of the structures by adjusting geometric parameters of inner and outer regions.The large effective refractive index difference between core and cladding region strongly affects the chromatic dispersion.The symmetry of structures prevents the existence of a linear electro-optic effect.
Fig. 1 shows the cross section of the four different structures along with their geometric parameters for better understanding.For outer structure N denotes the number of air-hole rings and r is the radius air-hole.If outer structure is hexagon, then a represents pitch and if outer structure is circular, then d represents the radial distance of the circular structure.For inner structure N1 represents number of airhole rings and r1 is the radius of air-hole.If inner structure is hexagon, then a1 represents pitch and if inner structure is circular, then d1 represents the radial distance between circular rings.The size of the The wavelength dependent refractive index of the silica was included in the simulation from Sell Meier equation.Chromatic dispersion is the phenomenon in which the phase velocity of light wave depends on its frequency.Dispersion D in ps/nm/km of a PCF is evaluated by calculating   value vs. the wavelength using the eq. 1 where c is the velocity of light in vacuum,  is wavelength and [  ] is the real part of the refractive index.Chromatic dispersion D is also dependent on the geometric parameters like shape of air holes, pitch, and inner circular holes diameter.By optimizing these parameters, suitable guiding properties can be obtained.
Confinement loss   is the light confinement ability within the core region and is evaluated using eq. 2. The increase of number of air hole rings support the confinement of light in the core region, which results in minimum losses than those with less air hole rings.Also, increasing the air holes radius results in the increasing of the air filling fraction and accordingly decreasing the loss.
where  0 = 2/ is wave number in free space with the unit dB/m, [  ] is the imaginary part of the refractive index.From eq. 1 and 2, it is observed that the dispersion is dependent on real value of refractive index and confinement loss is dependent on imaginary part of refractive index with wavelength.Eq. 3 below shows the relation between refractive index and group velocity.
where   is the refractive index and   is the group index.The wave number k can be considered as the change in spectral phase per unit length [10].
The Effective modal area Aeff was calculated using equation 4 The effective core radius is 0.4×a.In particular, the effective nonlinearity of the fiber can be calculated in the core of the fiber.The effective nonlinear coefficient is calculated by equation 5.
where  2 is material nonlinear coefficient.In this paper  2 used as silica fiber value is 2.6×10 -20 m 2 W -1 which determined by fiber designs.
A Finite Difference Time Domain (FDTD) Lumerical solution simulation tool was used to evaluate the performance of these structures.The simulation region was trimmed by applying a perfectly matched layer surrounding the structure.

III. RESULTS AND ANALYSIS
It has been an attempt by every research scholar in the field of fiber optics communication, to produce an optimized PCF structure which enables better negative dispersion and near zero confinement loss.In the proposed four different structures, changes in various combinations of PCF parameters were carried out during simulation till the optimized structure is found.The objective was to achieve better results than the results produced by other research scholars so far.
From fabrication perspective ± 2% manufacturing tolerance is generally considered [11].Thus each of the structure's geometric parameters were varied up to ± 2% and simulation was carried out.
Finally, all structures (without considering any manufacturing inaccuracies) were compared with each other for better dispersion and nearly zero confinement loss.In fig.2, C-H structure shows the effect of the ± 2% variation in the parameter of r1 and d1 of inner region and r and a of outer region on dispersion.For this study, changes were done to only one of the parameters keeping remaining parameters at optimum (nominal) value.For example, variation of ± 2% in optimum value of pitch which is 1.8 µm considered while keeping other nominal dimensions constant to r = 0.7 µm, d1 = 0.21 µm and r1= 0.1 µm.Similar study was carried out for other three structures also.A ± 2% variation in r results in ± 3% change in average dispersion, while ∆D increases more than 100%, whereas ± 2% variation in a results the average dispersion by ~ ± 2% with around 50% increases in ∆D.Keeping outer parameter at optimum value and varying inner parameter by ± 2% in r1 results in ± 4% changes in average dispersion with ~60% increase in ∆D.And ± 2% variation in d1 results in ± 3% changes in average dispersion with ~80% increase in ∆D.
The variation in these four parameters (one at a time) do not change the shape of the dispersion curve significantly.There is small variation in guided mode profile corresponding to the changes in one of the parameters at any given instance.Similar observation was registered for other three structures also.
In fig.3, H-C structure demonstrates sensitivity of dispersion for ± 2% variation in the parameter of r1, a1 of inner region and r, d of outer region.From the curve it is seen that ± 2% variation in r1 results in ~ ± 3% changes in average dispersion with 60% increase in ∆D.Similarly, ± 2% variation in a1 results in ~ ± 3% changes in average dispersion with 60% increase in ∆D and ± 2% variation in outer region r, d parameters affects ± 4% and ± 3% variation in average dispersion respectively with 80% and 60% increase in ∆D respectively In fig.4, H-H structure shows the changes in average dispersion on changing inner (r1, a1) and outer parameter (r, a).± 2% variation in r1, a1 results in ± 2% and ± 4% changes in average dispersion respectively whereas ∆D increases more than 70% and higher than100 % respectively.± 2% variation in r, a results in ± 5% and ± 8% changes in average dispersion respectively whereas ∆D increases more than 60% and more than 85 % respectively.
In fig. 5, C-C structure shows the changes in average dispersion on changing inner (r1, d1) and outer parameter (r, d).± 2% variation in r1, d1 results in ± 3% and ± 2% changes in average dispersion respectively while ∆D increases more than 70% and more than 60 % respectively.± 2% In general, while the pitch a or d of a PCF decreases, the negative dispersion increases whereas increase in r results in increase in negative dispersion which suggests reduction in effective refractive index.
Effective mode area represents quantitative measurement of the area covered by the fiber's guided mode.Fig. 6, 7, 8 and 9 demonstrate that effective mode area increase with an increase in operating wavelength and corresponding nonlinear coefficient γ decreases with an increase in operating wavelength.The calculated nonlinear coefficient of all four proposed structures C-H, H-H, H-C, C-C are 11, 17.6, 18.67, 25.08 W -1 Km -1 at 1.55 µm respectively with material related nonlinear coefficient is 2.6 × 10 -20 m 2 W -1 .Four-wave mixing is considered to be harmful for multi channels fiber optic communication system such as WDM, therefore lower nonlinear coefficient from transmission medium is required to reduce four-wave mixing nonlinear effect.Of all the four structures, structure C-H exhibit smaller nonlinear coefficient and high effective mode area as compared to remaining three structures.any given ratio of r1/d1.Also there is increase in effective mode area for any given value of r/a ratio as the ratio r1/d1 increases.This is due to the fact that the average effective refractive index at core region reduces with respect to increase in r1.Hence light confinement in core region is weaker which means it is forced out to the cladding region by inner array of circular holes.This also means that light confinement is better around the core region as the ratio r/a increases.The absolute birefringence of all these structures are ~10 -8 at 1.55 µm wavelength.This is negligible value due to symmetric structures, hence the effective index of the x-polarized and ypolarized guided mode is degenerated [13].By varying geometrical parameters of the C-H structure, broadband PCF with large negative dispersion over 350 nm range and very low confinement loss can be obtained.Besides comparing with previous complicated structures, the design procedure of C-H structure is much easier as very few parameters need to be optimized.We believe that the proposed C-H structure can be used for future high speed transmission.

Fig. 2 ,
Fig. 2, 3, 4 and 5 shows the changes in the dispersion values against the selected wavelength range for variation in geometric parameters by ± 2% for all PCF structure models Circle in Circle (C-C), Hexagon in Hexagon (H-H), Circle in Hexagon (C-H) and Hexagon in Circle (H-C) presented.However, one should note that after fabrication, geometric parameters of the cross section does not vary in only one direction i.e. either +2% or -2%.They are normally mixed and therefore the final effect is more or less balanced.Therefore, more weightage should be given to the results carried out for the nominal dimensions of the geometric parameters.Fig.2, 3, 4 and 5 shows dispersion values for optimized parameters and changing one of the parameters while keeping other parameters constant.Structure C-H shows higher negative dispersion range values as compared to other structures considering ± 2% variation in the geometrical parameters.

Fig. 12 Fig. 11 .Fig. 12 .
Fig. 12 compares the effective mode area against r1/a1 ratio in case of inner hexagon structure and r1/d1 in case of inner circular structure at wavelength 1.55 µm.Structure C-H shows larger increase in effective mode area corresponding to increase in r1/a1 or r1/d1 ratio as compared to other three

Fig. 14 compares
Fig. 14 compares real and imaginary part of effective refractive index against wavelength for all four different structures.Real part of refractive index is shown by solid lines whereas imaginary part of refractive index is shown by dashed lines.Structure C-H shows higher effective refractive index than the other remaining structures.Linear form of the real part of the refractive index shows that the light is confined and non-dispersive in nature.Imaginary part of refractive index is nearly zero which leads to near zero confinement loss based on equation 2.

Fig. 15 showsFig. 15 .
Fig.15 shows the timeaveraged poynting vector profile for two degenerate modes at 1.55 µm in propagation direction.Modes profiles of C-C and H-C structures show that the light propagate towards inner structure, whereas H-H structure shows the less light intensity.It can be observed that the modes profile of C-H structure is more confined around outer region of PCF core.This is due to relatively high refractive index at outer region (cladding) as compared to lower refractive index of µm to 1.7 µm and the Design and Modal Analysis of Photonic Crystal Fiber for Dispersion Compensation over Broadband Range Madhavi Waghmare 1 , K.T.V.Reddy 2 , design has been studied with Ge doped core by using genetic algorithm.Asiful Islam et.al. [7] used equiangular spiral PCF structure and achieved absolute dispersion variation of -227ps/nm-km with ∆D ~ 11 ps/nm-km over frequency range 178.98 THz to 202.56 THz.Tee et.al. [8] used PCF in PCF structure and achieved residual dispersion compensation -457.4 ps/nm/km over the range larger than E to U band i.e. (177.3THz to 220 THz) and ∆D ~11.9 ps/nm/km.All these studies were carried out on a 2D PCF with uniform air hole structures for high speed and broadband transmission.However, there is still a possibility to achieve more negative dispersion and/or near zero confinement loss over broadband optical frequency range.Higher flattened negative dispersion indicates good dispersion compensation and almost zero confinement loss resulting in enhancement of fiber bandwidth.In this paper we explained the investigations of different new PCF structures which can provide very large negative dispersion.Four new PCF structures are designed to compensate dispersion of [9]pagation of light with nearly zero confinement loss.In these structures, inner circular air hole array with different geometry is used to reduce dispersion.In this paper we implemented and studied the effect of the defect introduction in the newly designed PCF structures within transmission spectrum range from 190.95 THz to 245.73 THz i.e. 1.22 µm to 1.57µm with -704.62 (ps/nm/km) average large negative dispersion.II.MODELLING OF A PHOTONIC CRYSTAL FIBER STRUCTUREPCF can have air hole structure of any shape.Commonly two different forms of lattice are used.One with square lattice and other with triangular lattice.PCF has gathered more attention because of their special characteristics in triangular lattice of air holes[9].Research reports have studied Journal of Microwaves, Optoelectronics and Electromagnetic Applications, Vol.15, No. 4, December 2016

Table II
shows comparison of dispersion, effective mode area and nonlinear coefficient obtained earlier by different research scholars.

TABLE II SUMMARY
OF THE PREVIOUS RESULTS Refractive Index (Re) v/s Wavelength ComparisonTable III below shows the range of the dispersion, nonlinear coefficient and Effective mode area obtained in this research for each geometric structure for fixed wavelength range 1.22 µm to 1.57 µm.Dispersion range values shown are based on consideration of ± 2% variation in the geometrical parameters.Effective mode area and nonlinear coefficient values are calculated for nominal dimensions (without consideration of ± 2% variation) of geometric parameters.Structure C-H is able to handle wavelength band O+E+S+C and beyond up to certain extent on O band side.
Table III list of the different structures and their parameters