Comparative Study of Fundamental Properties of Honey Comb Photonic Crystal Fiber at 1 . 55 μ m Wavelength

Fundamental properties such as mode field distribution, real effective refractive index, imaginary effective refractive index, confinement loss of two new kinds of honeycomb photonic crystal fibers are successfully studied by using Full-Vectorial Finite element method (FV-FEM). Low confinement loss 0.1×10dB/km is achieved at wavelength 1.55μm in hollow core honey comb PCF by removing 6-air holes in cladding region with air hole diameter 1.38μm in cladding region, pitch 2.3μm and air core diameter 0.2μm.


I. INTRODUCTION
Recently, Photonic Crystal fibers (PCFs) have diverse applications in supercontinuum generation, nonlinear optics, telecommunications, sensors, soliton, lasers, medical instrumentations etc [1][2][3][4][5].Photonic crystal fibers can be divided into two modes of operation, according to their mechanism for confinement.Those, with a solid core or a core with a higher average index than the microstructured cladding, can operate on the same guiding principle as conventional optical fiber called as solid core PCF [6].Alternatively, one can create a defect by introducing an air hole in core region which is also known as hollow core PCF where light can confine in a lower-index core and even a hollow (air) core and light is confined by a photonic band gap effect.Potential advantage of a hollow core is that one can dynamically introduce materials into the core, such as a gas that is to be analyzed for the presence of some substance.However, they can have a much higher effective-index contrast between core and cladding and therefore can have much stronger confinement for applications in nonlinear optical devices, polarization-maintaining fibers.In particular, when the defect is formed by removing several air holes in cladding region of PCF the structure becomes honeycomb PCF.If we introduce a small air core as defect in core region of honeycomb PCF we call it as hollow core or air-guiding PCF.In this fiber, light guiding mechanism is due to completely photonic band gap effect [7].For designing Photonic band gap fibers (PBGFs),

Comparative Study of Fundamental
Properties of Honey Comb Photonic Crystal Fiber at 1.55µm Wavelength 344 not only the core structure but also the cladding structure is very important.When large band-gaps are obtained in the cladding, the fiber can possess desirable properties such as a low confinement loss and a broadband transmission range.Generally, a triangular lattice for the cladding structure is used for realizing airguiding PBGFs because of its large band-gaps.So a honeycomb structures for air-guiding PBGFs with large band-gaps have also been proposed [8,9] and have gathered much attention.In addition, a modified honeycomb structure has been investigated by Broeng et al. [10].However it has been evaluated only how the presences of interstitial air holes affect the bandgap properties.More recently, Chen et al suggested the possibility that the modified honeycomb structure produces completely different large band-gaps beyond the realm of triangular and honeycomb lattices as structural parameters are adjusted [11].Saitoh et al [12] and Vincetti et al [13] calculated a modal dispersion curve and confinement loss properties for modified honey comb type air guiding PBGFs with large core diameters, which suffers from multi-mode transmission.However, the possibility of realizing single-mode air-guiding PBGFs operating in a wide wavelength range with low confinement losses has not been investigated so far.Confinement loss is the important issues in optical communication through optical fiber [14].
Recently, different kinds of PCFs have been designed by taking appropriate geometrical parameters such as air hole diameter, pitch and air core diameter with various propagation properties e.g.dispersion, birefringence, confinement loss, nonlinearity etc [15,16].Confinement loss in conventional optical fiber is very high.To overcome these limitations of conventional optical fiber an alternate fiber e.g.hollow core honey comb photonic crystal fiber is developed.Due to complex structure of PCF, different numerical techniques has been used by researchers to study the different properties of PCF such as Finite Element Method (FEM) by Brechet et.al in 2000 [17], Multipole Method (MM) by White et.al [18], Finite Difference Time Domain Method (FDTDM) by Qiu et.al [19], Plane Wave Expansion Method (PWEM) by Guo et.al [20], Effective index Method by Chiang et.al [21], Vector wave Expansion method by Issa et.al [22], Time domain beam propagation method by Koshiba et.al [23] and Localized function Method by Mogilevtsev et.al [24] and other numerical modeling.
Finite element methods are divided into two catagory one is Scalar FEM and another is Full-vectorial FEM.
Scalar FEM technique is used only for solving nodal element but it fails to solve edge element in the domain where as FV-FEM technique is used to solve both nodal element as well as edge element in the domain.Here we have used FV-FEM, which is suitable for such analysis as it can handle complicated structure geometries and also takes less computational time.Two new types of hexagonal triangular-based cladding structure PCF are used, where refractive index of air and silica are 1 and 1.45 respectively.By manipulating circular air hole diameter 'd', pitch 'Λ' and air core diameter d c , it is possible to control the properties of PCF such as real effective refractive index, imaginary effective refractive index, confinement loss, at wavelength 1.55µm.

II. FULL-VECTORIAL FINITE ELEMENT METHOD
The Full Vectorial Finite element method (FEM) is generally advantageous in complex geometries of photonic crystal fiber.It is a full vector implementation for both propagation and leaky modes and cavity modes for two dimensional Cartesian cross sections in cylindrical co-ordinates.First and second order interpolant basis are provided for each triangular elements.PEC (Perfect electrical conductor) or PML (perfectly matched layer) boundary conditions is employed at computational domain for evaluating total dispersion and confinement loss of proposed PCF [25].
Under the source free condition time dependent Maxwell's curl equation can be expressed a Where ω is the angular frequency,ߤ and ℇ are the permeability and permittivity of free space, and ሾߤ ሿ and ሾℇ ሿ are, respectively the relative permeability and permittivity tensors of the medium given by From Equation ( 1) and ( 2), we can derive the vectorial wave equation as Where ݇ = ߱ඥμ ℇ is the wave number in free space, ϕ is either the electric field E or the magnetic field H, and the tensor ሾ‫‬ሿ and ሾ‫ݍ‬ሿ are given by For ϕ=E and For ϕ=H In this chapter we have considered 2-dimension photonic crystal fiber with triangular lattice which are uniform along the z direction and periodic in the x-y plane.The field distribution ϕ of the wave modes in plane propagation can be expressed as Where ߶ ௧ and ߶ ௭ , both assumed to be function of x and y, are the transverse and longitudinal field components of ϕ, respectively.Substituting Eq. ( 10) into Eq.( 5) and using

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Where ߘ ௧ and ߘ ௭ are, respectively, the transverse and longitudinal parts of ∇ operator.Eq.( 8) can be separated into its transverse component And its longitudinal component For hybrid node/edge FEM [26], the transverse components are expanded in a vector (edge element) basis.
Where, E Ti are the values of the field along each edge.The longitudinal component (perpendicular to the plane of the element) is represented by a scalar (node element) basis.
Where, E zi are the values of the field and N i are the basis at each node.The basis dimension 'n' depends on the geometry of the element and the order of the interpolation.Since the Euler Langrangian equations of the functional correspond to original wave equations the solution of latter equations can be approximated by extremization of the functional [27][28][29][30][31].The functional are approximated using interpolation of polynomial basis functions and functional are discretized in a finite no. of element within the computational domain.
Finally, we will get matrix generalized eigen-value equation of the form  By using FV-FEM simulation technique [32], first we calculated real effective refractive index with wavelength range 0.9 µm to 1.7 µm for first kind of PCFs as shown in Fig. 1(a).We observed that real part effective refractive index decreases with increasing air hole diameter as well as increasing wavelength and it is found that real part effective refractive index is 1.413756 at 1.55 wavelength when air hole diameter d=1.38µm as shown in Fig. 2(a).For second kind of PCFs, it is found that real part effective refractive index decreases with increasing wavelength and increases with increasing air core diameter and real part effective refractive index is obtained 1.410566 at 1.55µm wavelength when air core diameter is d c =0.2µm as shown in Fig. 2(b).The transverse electric mode field pattern at wavelength 1.55µm of two kinds of PCFs are shown in Fig. 3(a    Fig. 4(a) Variation of Real part effective index with normalized frequency of Solid core honeycomb PCF having air hole diameter d=1.38µm,1.36µmand 1.34µm of same pitch Λ=2.3µm for all three structures.Fig. 6(a) Variation of confinement loss with wavelength of Solid core honeycomb PCF having identical air hole diameter d=1.38µm,1.36µmand 1.34µm of same pitch Λ=2.3µm for all three structures.

III. RESULTS AND DISCUSSION
17) Where A and B represent global finite matrices, E Ti are Transeverse electric field and n eff represent the modal effective refractive index, n eff =β/k 0 .Here β is the propagation constant for guided mode and k 0 is the propagation constant for free space, k 0 =2π/λ.

Fig. 1 (
Fig. 1(a) Solid Core Honey Comb PCF ) and 3(b) respectively.It shows that mode field pattern is confined more in second kind of PCFs in comparison to 1 st kind.

Fig. 2 (
Fig. 2(b) Variation of Real part effective refractive index with wavelength of Hollow core honeycomb PCF having air core diameter d c =0.2µm,0.1µm,and0.05µm of same pitch Λ=2.3µm and air hole diameter d=1.38µm for all three structures.

Fig. 4 (
Fig. 4(b) Variation of Real part effective index with normalized frequency of Hollow core honey comb PCF having air core diameter d c =0.2µm,0.1µm,and0.05µm of same pitch Λ=2.3µm and air hole diameter d=1.38µm for all three structures.As clear from Fig.4(a) and 4(b), the real part effective refractive index increases with increasing normalized frequency and decreases with decreasing air hole diameter in solid core PCF and air core diameter in hollow core PCFs.

Fig. 5 (Fig. 5 (
Fig.5(a) Variation of Imaginary part effective refractive index with wavelength of Solid core honeycomb PCF having identical air hole diameter d=1.38µm,1.36µm,and1.34µm of same pitch Λ=2.3µm for all three structures.

Fig. 6
Fig.6 (b) Variation of confinement loss with wavelength of Hollow core honeycomb PCF having air core diameter d c =0.2µm, 0.1µm and 0.05µm of same pitch Λ=2.3µm and air hole diameter d=1.38µm for all three structures.
Variation of Real part effective refractive index with wavelength of Solid core honeycomb PCF having identical air hole diameter d=1.38µm,1.36µm,and1.34µm of same pitch Λ=2.3µm for all three structures.