On-line version ISSN 2179-1074
J. Microw. Optoelectron. Electromagn. Appl. vol.10 no.2 São Caetano do Sul Dec. 2011
Comparative study of fundamental properties of honey comb photonic crystal fiber at 1.55µm wavelength
S.S. MishraI; Vinod Kumar SinghII
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 0.1×10-4dB/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.
Index Terms: Honey comb Photonic crystal fiber, Full-Vectorial Finite Element Method, Effective refractive index, modal field pattern, confinement loss.
Recently, Photonic Crystal fibers (PCFs) have diverse applications in supercontinuum generation, nonlinear optics, telecommunications, sensors, soliton, lasers, medical instrumentations etc [1-5]. 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. 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. For designing Photonic band gap fibers (PBGFs), 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.. 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 . Saitoh et al  and Vincetti et al  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 . 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 , Multipole Method (MM) by White et.al , Finite Difference Time Domain Method (FDTDM) by Qiu et.al , Plane Wave Expansion Method (PWEM) by Guo et.al , Effective index Method by Chiang et.al , Vector wave Expansion method by Issa et.al , Time domain beam propagation method by Koshiba et.al  and Localized function Method by Mogilevtsev et.al  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 dc , 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 .
Under the source free condition time dependent Maxwell's curl equation can be expressed a
Where ω is the angular frequency, µ0 and ε0 are the permeability and permittivity of free space, and [µr] and [εr] 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 is the wave number in free space, ϕ is either the electric field E or the magnetic field H, and the tensor [p] and [q] are given by
For ϕ=E and
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 ϕt and ϕz, 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
Where ∇t and ∇z 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 , the transverse components are expanded in a vector (edge element) basis.
Where, ETi 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, Ezi are the values of the field and Ni 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 [27-31]. 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
Where A and B represent global finite matrices, ETi are Transeverse electric field and neff represent the modal effective refractive index, neff=β/k0. Here β is the propagation constant for guided mode and k0 is the propagation constant for free space, k0=2π/λ.
III. RESULTS AND DISCUSSION
We have compared fundamental properties of such as mode field distribution, real part and imaginary part of effective refractive index, dispersion and confinement loss properties of two new kinds of honeycomb PCF by using FV-FEM. In our designing for first kind PCFs, 6-air holes are missing in cladding region and one air hole in core region (Solid Core Honey comb PCF) and in another type only 6-air holes are missing in cladding region (Hollow Core Honey Comb PCF). Black regions are air hole and white regions are solid silica as shown in fig 1(a) and 1(b) respectively. For the first kinds of honey comb fiber we have varied air hole diameter d=1.34 µm, 1.36 µm and 1.38 µm respectively for three structures by keeping pitch (2.3 µm) same. For the second kinds of honey comb fiber we have varied air core diameter dc as 0.05µm, 0.1µm and 0.2µm respectively by taking identical air holes in cladding region and pitch equals to d=1.38 µm and Λ=2.3 µm respectively.
By using FV-FEM simulation technique , 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 dc=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) and 3(b) respectively. It shows that mode field pattern is confined more in second kind of PCFs in comparison to 1st kind.
The variation of effective refractive index with normalized frequency was also studied for different fiber parameter in both cases. 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.
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.
Imaginary effective refractive index is an important parameter for analysis confinement loss of PCFs. We calculated imaginary part of effective refractive index corresponding to the same wavelength range for two kinds of PCFs under studied. It is observed that imaginary part of effective refractive index is decreases with increasing air hole diameter and air core diameter. It is also increases with increasing wavelength for all three structures. Imaginary effective refractive index at wavelength 1.55µm is obtained 2.176×10-5 when the air hole diameter is 1.38 for first kind PCF and 1.386×10-5 is obtained when air core diameter is 0.2µm as shown in Fig. 5(a) and 5(b) respectively.
After getting imaginary effective refractive index we can easily calculate confinement loss of two kinds of PCFs by using following formula given bellow.
Where Im[neff] represented as imaginary effective refractive index and k0 =2π/λ, with λ is the propagation wavelength.Once we got imaginary effective refractive index, we can calculate confinement loss of PCF easily by using the equation (35). It is found that confinement loss decreases with increasing air hole diameter and increases with wavelength for two kinds of PCFs as shown in Fig.6(a) and 6(b). It is clear that at wavelength 1.55µm confinement loss is very-very small e.g 0.1×10-4dB/km in hollow core honeycomb PCF in comparison to solid core honey comb PCF.
Fundamental properties of two new kinds of Honey comb PCF are studied successfully via FV-FEM simulation technique. The simulation result proves that air holes of the cladding and air core in core region have a strong influence on the PCFs to confinement loss. Hollow core honey comb PCFs having identical air hole diameter d=1.38µm, air core diameter dc=0.2µm and pitch Λ=2.3µm shows very small confinement loss e.g 0.1×10-4dB/km at wavelength 1.55µm. This kind of PCFs may be used in telecommunication as well as gas sensing purpose.
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Received 11 May 2011; for review 13 May 2011; accepted 23 Sept. 2011