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1 Abstract-- In this paper, we have modeled a solid-core photonic crystal fiber (PCF) by using full vectorial finite element method (FEM). Modal properties of the PCF have been investigated by calculating the effective index of the supported mode. Results for effective index and the dispersion are presented. COMSOL Mul-tiphysics software has been used to calculate the effective index of the supported mode of the fiber, both with and without includ-ing fundamental space-filling mode (FSM) layer in the fiber de-sign. Hexagonal geometry with three ring structure is used to model the fiber; however, the present analysis can also be applied to any kind of fiber geometry.

Index Terms—Finite element method (FEM), fundamental space-filling mode (FSM), photonic crystal fiber (PCF), solid-core photonic crystal fiber (SC-PCF), waveguide dispersion

I. INTRODUCTION HOTONIC crystal fibers (PCFs) are a new class of optical fibers which have the potential to revolutionize optical

fiber technology. Apart from the term photonic crystal fiber, these fibers are also referred to by some other names in the literature such as holey fibers (HF) and microstructured opti-cal fibers (MOF). These fibers are essentially low-loss wave-guides and consist of a core which is surrounded by a periodic array of air holes in the cladding region. This configuration has led to a number of novel properties like endless single mode (ESM) operation, controllable birefringence and disper-sion characteristics and a high degree of nonlinearity [1]. Due to these properties, it has potential applications in high power fiber lasers, fiber amplifiers, non-linear devices, optical sen-sors, fiber-optic communication and many others [2].

The cladding of a photonic crystal fiber can be constructed with a structure similar to that found in photonic crystal. This is where the term ‘photonic crystal fiber’ originates. Photonic crystals, first studied by Yablonovitch [3] and John [4] are

A. G. Prabhakar is with the Department of Electrical Engineering, Delhi Technological University, Delhi-110042, India (e-mail: gautamprabha-kar.gp@gmail.com). B. A. Peer is with the Department of Computer Engineering, Delhi Technolo-gical University, Delhi-110042, India (e-mail: akshit.peer@gmail.com). C. A. Kumar is with the Department of Applied Physics, Delhi Technological University, Delhi-110042, India (e-mail: ajeet.phy@dce.edu). D. V. Rastogi is with the Department of Physics, Indian Institute of Technol-ogy Roorkee, Roorkee-247667, India (e-mail: vipul.rastogi@osamember.org).

essentially a photonic analog of the electronic crystal. They are periodic structures on the scale of optical wavelength (much larger than the atomic size of electronic wavelengths). This leads to Bragg-like diffraction resulting in photonic bandgaps, which are a range of frequencies for which light cannot propagate through the crystal [5].

There are two classes of photonic crystal fibers viz. solid-core PCF (SC-PCF) and hollow-core PCF (HC-PCF). The term solid-core photonic crystal fiber refers to those structures that have a solid core which is usually made of silica. These fibers guide light by the phenomenon of total internal reflec-tion. On the other hand, hollow-core photonic crystal fiber has an air hole in the core region and transmits light by photonic bandgap type guidance [6]. In this paper, we will be dealing with SC-PCF and henceforth, HC-PCF will not be considered.

A solid-core photonic crystal fiber is an all-silica fiber con-sisting of a solid core surrounded by an array of air holes in the cladding. The array of air holes forms a medium with an effective refractive index below than that of the core region. This results in a refractive index profile that is quite similar to that of step index fiber, with the array of air holes becoming an effective cladding to the high index core region. It is for this reason that these fibers guide the light by the same phe-nomenon as the step index fiber does i.e. by total internal ref-lection [7]. The air holes lower the effective refractive index of the cladding region to levels much lower than that of stan-dard optical fiber. This results in a large numerical aperture (NA) for the structure i.e. a very strong guidance effect.

In this paper, we have modeled a solid core PCF using COMSOL Multiphysics finite element method (FEM). For designing the fiber, we have used hexagonally symmetric SC-PCF. We have calculated the effective index and dispersion loss of the fiber both with and without including fundamental space-filling mode (FSM) layer in the fiber design. The results have been compared with those present in the literature and are shown to be in good agreement with them.

II. FIBER DESIGN

We have designed a solid-core PCF having a hexagonal geometry with a periodic triangular lattice arrangement of air holes such that the holes are arranged at the corners of an equilateral triangle [8]. This type of arrangement can be de-

Finite Element Analysis of Solid-Core Photonic Crystal Fiber

A. Gautam Prabhakar, B. Akshit Peer, C. Ajeet Kumar and D. Vipul Rastogi

P

scribed by two parameters: pitch Λ (i.e. periodic length be-tween the holes) and filling factor /d Λ (i.e. the ratio of hole size to pitch). The properties of the PCF can be controlled to a large extent by varying these parameters. The degrees of free-dom available in designing pitch and filling factor result in a number of interesting properties of PCF.

In the fiber, the core is surrounded by 3 successive rings of such air holes with radii Λ , 3 Λ and 2 Λ respectively, each containing six holes placed symmetrically on the ring, as shown in Fig. 1. Generally there are several layers of holes in a typical fiber. However, only the first few of these signifi-cantly change the optical properties of a fiber. Thus, we con-sider only three layers of holes for the present model.

Fig. 1. Geometry of solid-core photonic crystal fiber.

In this ring picture of the fiber, the holes are regularly ar-ranged around the core and the rings have a six-fold angular symmetry, with Λ as the pitch and d as the hole diameter. Thus, the geometry of fiber is such that the axis θ = 0 (accord-ing to polar coordinates) passes through the center of the fiber. Inclination of holes in the 1st and 3rd rings is same while the holes in 2nd ring are shifted by an angle of π/6 [8].

In the second approach, a uniform medium of index equal to the effective index of the fundamental space-filling mode (FSM) beyond the holes of the third ring has been used, as shown in Fig. 2. The fundamental space-filling mode of a PCF is defined as the mode with the largest modal index of the infinite two-dimensional photonic crystal that constitutes the PCF cladding [9].

Fig. 2. Geometry of solid-core photonic crystal fiber with FSM layer.

III. MODELING AND ANALYSIS

A. Modeling To model the fiber in COMSOL, following steps are fol-

lowed in order (Fig. 3):

Fig. 3. Flowchart for modeling of PCF in COMSOL.

In geometrical modeling, a solid-core PCF having hexago-nal geometry with appropriate parameters is designed. In the next step, the physical parameters such as wavelength of light used and refractive index of silica and air holes are specified using the Sellmeier’s equation. This is followed by generation of triangular mesh and its further refinement. The effective index of fiber is thus computed by solving the eigenvalue equation for each of these triangular meshes using COMSOL’s FEM. Finally, the results are interpreted using post-processing and visualization tools [10]. This includes generation of various graphs and field plots.

The analysis of optical waveguides is based on Maxwell’s equations. For our model of optical fiber, we will assume that the fiber consists entirely of homogenous dielectric material. Also, it is assumed that no sources of light exist inside the fiber, so that no free charges or currents exist inside the fiber and thus ρ and J are zero in Maxwell's equations.

We further assume that there is a linear relationship be-tween electric field (E) and displacement (D) and the mate-rials within the fiber (typically silica and air) are macroscopic and uniform in all directions. This ensures that E and D are related by a scalar dielectric constant ε. Last it is assumed that the fiber is lossless and thus dielectrics are described by real numbers [6]. Using these assumptions, we obtain the eigenvalue equation:

2

E Ecω ε⎛ ⎞∇×∇× = ⎜ ⎟⎝ ⎠

B. Finite Element Method The finite element method (FEM) is a numerical tech-

nique for finding approximate solutions of partial differential equations as well as integral equations. This method approx-imates the PDE as a system of ordinary differential equations which can be solved separately using various numerical tech-niques. While solving the eigenvalue equation (1), COMSOL

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Multiphysics uses the finite element method. It sub-divides the object into very small but finite size elements. This process is called ‘meshing’ and is shown in Fig. 4 for the sol-id-core PCF. Each element of the mesh is governed by a set of characteristic equations which describe its physical properties and boundary conditions. These equations are then solved as a set of simultaneous equations to compute the effective index of the modes supported by the fiber [10].

Fig. 4. Solid-core PCF with triangular meshing.

IV. SIMULATIONS AND RESULTS The COMSOL’s finite element method (FEM) has been

used for the modal analysis of a solid-core photonic crystal fiber formed by three ring structure with a hexagonal ar-rangement of circular air holes on the cladding of pure silica. The results obtained have been compared with those obtained using other methods available in the literature [2]. In the simu-lations, the refractive index of silica n(λ) has been obtained using Sellmeier’s equation [11] given below:

2 4 30 1 2 2

542 2 2 3

( )( 1)

( 1) ( 1)

Cn C C C

CC

λ λ λλ

λ λ

= + + + +−

+− −

where C0 = 1.4508554, C1 = − 0.0031268, C2 = − 0.0000381, C3 = 0.0030270, C4 = − 0.0000779, C5 = 0.0000018, l = 0.035 and is measured in μm.

For the simulations, the core radius is taken as 0.625 Λ . All the calculations are done for Λ = 3μm, and /d Λ = 0.2. For this optical fiber, the plot of time average power flow in z direction (Fig. 6) is obtained using our model for λ = 1.50 μm.

The numerical results for effective index and dispersion of the solid-core PCF defined above have been tabulated in Ta-ble I and Table II.

In Table I, the refractive index of silica is calculated using (2). The effective index of the fiber neff is calculated and com-pared with a reference effective index taken from the literature [2] and the percentage error is calculated. The dispersion in

the fiber is calculated using (4) and the variation of dispersion with wavelength is plotted in Fig. 8.

In Table II, the refractive index of the FSM layer is calcu-lated by scalar analytical approach (SAA) using (3). In this approach, a hexagonal unit cell is approximated as a circular one as shown in Fig. 5.

Fig. 5. Approximation of a hexagonal unit cell for a circular one in SC-PCF.

Fig. 6. Field plot of time average power flow (W/m2) (a) without FSM and (b) with FSM.

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(b)

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TABLE I EFFECTIVE INDEX OF THE FUNDAMENTAL MODE OF SOLID-CORE PCF (WITHOUT FSM) AT VARIOUS WAVELENGTHS

S.No. Wavelength (µm)

Refractive Index (Silica) Reference neff [2]

neff using COMSOL's

FEM

Percentage Error in neff

(x10-2)

Dispersion (ps/km-nm)

1. 1.50 1.4450 1.437978 1.437371 4.22298 17.25

2. 1.20 1.4484 1.443370 1.443075 2.04424 09.40

3. 1.10 1.4495 1.445168 1.444902 1.84095 -25.94

4. 1.00 1.4507 1.447012 1.446800 1.46530 -38.08

5. 0.60 1.4584 1.456844 1.456866 0.15100 -267.20

6. 0.55 1.4602 1.458943 1.458899 0.30159 -427.90

7. 0.50 1.4626 1.461572 1.461516 0.38316 -605.00

TABLE II

EFFECTIVE INDEX OF THE FUNDAMENTAL MODE OF SOLID-CORE PCF (WITH FSM) AT VARIOUS WAVELENGTHS

S.No. Wavelength (µm)

Refractive Index (Silica)

Refractive Index of FSM Layer Reference neff [2]

neff using COMSOL's

FEM

Percentage Error in neff

(x10-2) 1. 1.50 1.4450 1.4424 1.437179 1.437080 0.68884

2. 1.20 1.4484 1.4441 1.442771 1.443186 2.87640

3. 1.00 1.4507 1.4454 1.446564 1.446808 1.68675

The characteristic equation for SAA is:

0 00

( ) ( )( ) ( )( ) ( )

( )( ) ( )( )

l l

l

ll l

l

I wa J uRw J ua Y uaI wa Y uR

J uRu J ua Y uaY uR

⎡ ⎤−⎢ ⎥

⎣ ⎦

⎡ ⎤= − −⎢ ⎥

⎣ ⎦

In (3), 2 2 2 2 2 2 2 2, , / 2silica airu n k w n k a dβ β= − = − = is the air hole radius, R is the radius of the equivalent circular unit cell as shown in Fig. 5, Il is the modified Bessel function of the first kind of order l, and Jl and Yl are Bessel functions of the first kind and second kind of order l, respectively [9].

In Fig. 7, the variation of effective index as a function of wavelength is plotted for the solid-core PCF without FSM layer.

Fig. 7. Plot of effective index v/s wavelength.

One of the most significant parameter for an optical fiber is dispersion. For a photonic crystal fiber, effective control of dispersion characteristics can result in very high negative dis-persion as well as near zero dispersion over a broad range of wavelengths. However, it is important to calculate the effec-tive index of a fiber accurately as waveguide dispersion is related to the second order derivative of effective index with respect to wavelength [8]. Waveguide dispersion is given by:

2

2effd n

Dc dλ

λ= −

Using the above formula, waveguide dispersion is calcu-lated for solid-core PCF at different wavelengths and a graph is obtained using MATLAB. The graph is shown in Fig. 8.

Fig. 8. Dispersion characteristics of the PCF.

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(3)

V. CONCLUSION We have simulated a solid-core photonic crystal fiber with

three ring hexagonal geometry using finite element method. Numerical results of the effective index of the fiber are pre-sented and compared with those in the literature. Numerical simulations have shown that the results of the finite element method are in a very good agreement with those in the litera-ture.

Finite element method is one of the most widely used me-thods to model various problems due to its ability to handle complex calculations and analysis. It is a useful tool for com-puting the effective index of microstructured fibers which have complex geometries and structures.

However, it suffers from some inherent drawbacks. First, a general closed form solution of the effective index with varia-tion in the wavelength of light is not obtained. Second, power-ful computers with adequate memory and processing speed along with reliable FEM software are required to apply this method effectively. These are also essential to reduce the computation time of the effective index. Third, this method provides only approximate solutions to the problems.

Although the method was applied for a lossless fiber, its ex-tension to leaky structures is straightforward. It requires the inclusion of perfectly matched layer (PML) in the fiber design with suitable boundary conditions. Also, apart from the con-ventional hexagonal geometry used here, various other fiber designs like hollow core structures and those with elliptical holes can be analyzed using this method.

VI. ACKNOWLEDGMENT This work has been partially supported by the UK India Edu-cation and Research Initiative (UKIERI) major

award on “Application specific microstructured optical fi-bres.”

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[2] V. Jandieri, K. Yasumoto, A. Sharma and H. Chauhan , “Modal analysis of specific microstructured optical fibers using a model of layered cylindrical arrays of circular rods”, IEICE Trans. Electron., vol. E93.C, pp. 17-23, January 2010.

[3] E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics", Physical Review Letters, vol. 58, pp. 2059-2062, May 1987.

[4] S. John, "Strong localisation of photons in certain disordered dielectric superlattices", Physical Review Letters, vol. 58, pp. 2486-2489, June 1987.

[5] J.D. Joannopolous, P.R. Villeneuve and S. Fan, "Photonic Crystals: putting a new twist on light", Nature , vol. 386, pp. 143-149, March 1997.

[6] Stig E. Barkou Libori, “Photonic crystal fibers: from theory to practice”, Ph D Thesis, Technical University of Denmark, Kongens Lyngby, Denmark, February 2002.

[7] J.C. Knight, T.A. Birks, P.St.J Russel, and D.M. Atkin, “All-silica single-mode optical fibre with photonic crystal cladding”, Optics Letters, vol. 21, pp. 1547-1549, 1996; with errata vol. 22, pp. 484-485, 1997.

[8] Anurag Sharma and Hansa Chauhan, “A new analytical model for the field of microstructured optical fibers”, Opt. Quant. Electron., vol. 41, pp. 235-242, December 2009.

[9] Y. Li, C. Wang, Y. Chen, M. Hu, B. Liu and L. Chai, “Solution of the fundamental space-filling mode of photonic crystal fibers: numerical method versus analytical approaches”, Appl. Phys. B:Lasers and Optics, vol. 85, pp. 597–601, May 2006.

[10] COMSOL Multiphysics User’s Guide, Version: COMSOL 3.5, 2008 [11] Ajoy Ghatak, Optics, 4th ed., Tata McGraw Hill, New Delhi, 2009, pp.

10.1-10.15.