REPOSITÓRIO DA PRODUÇÃO CIENTIFICA E INTELECTUAL DA UNICAMP
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by Sociedade Brasileira de Microondas e Optoeletrônica e Sociedade Brasileira
de Eletromagnetismo. All rights reserved.
DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
Abstract—
A new design of an optical fiber coupler constituted by two and three cores based
on Step Index Holey Fiber (SIHF) is proposed and analyzed by an efficient Vectorial Beam
Propagation Method (VBPM) in conjunction with Genetic Algorithm (GA). The fiber is made
from pure silica and germanium doped silica and contains two or three identical cores with
index holey separated by a distance ℓ. The proposed structures exhibit a very simple
geometry, easy of fabrication, and each fiber that composes the coupler is obtained through
the inclusion of a very small air hole at the core of one conventional step index fiber. Firstly,
we consider the coupler constituted by two cores and the signal power launched in the left
core is completely transferred to neighbor core with 100 % of the coupling ratio. Next, we
consider the coupler constituted from three cores and the signal power launched in the central
core is transferred to neighboring cores with 50 % of the coupling ratio.
Index Terms— Step index holey fiber, optical coupling, genetic algorithm, finite elements and cylindrical perfectly matched layer.
I. INTRODUCTION
Recently, new types of optical fibers have attracted the attention of researchers all over the world due to important characteristics of transmission for short and large distances. In order to avoid power penalty due to nonlinear effects, a long-hall transmission system optical fibers with small but nonzero chromatic dispersion can be required [1]. In this work, coupling characteristics between the cores of a
SIHF are being evaluated. The
optical
fibers used in the coupler exhibit an ultra-flattened chromatic dispersion for a large bandwidth [2] and they have been optimized by a genetic algorithm.Generally, to implement the division process of optical power between the cores presented in a coupler, the most common approach is the fusion of the optical fibers together by keeping mutual contact in the fusion region. This fusion process results in a tapered region of fused optical material,
Step Index Holey Fiber Design by Genetic
Algorithm for Directional Coupling
José Patrocínio da Silva, João Maria Câmara,
Department of Electrical Engineering, Federal University of Rio Grande do Norte, UFRN, 59078970,
Natal-RN, Brazil, E-mail: [email protected], Federal University of Semiarid Region, UFERSA, [email protected]
59625-900, Mossoró-RN, Brazil
H. E. Hernández-Figueroa and V. F. Rodrigues-Esquerre
School of Electrical Engineering, University of Campinas-UNICAMP, 13082-970, Brazil, E-mail:
[email protected] and Department of Electrical Engineering, Federal University of Bahia, UFBA,
where the power transfer among fibers occurs due to their proximity [3]-[5]. Here, we analyze a novel and simple structure formed by two and three cores separates by a distance ℓ, constant along the propagation direction. It is important to point out that ultra-flattened dispersion chromatic is obtained by introducing a small air hole in the core of the optical fiber with conventional refractive index. The structure has been analyzed by using a computational code based in genetics algorithm applied in conjunction with an efficient vectorial finite element formulation with the inclusion of perfectly matched layer of cylindrical type [6]. Moreover, the finite element method is widely recognized as a powerful numerical tool for analysis of optical devices and in recent years has been largely reported in the literatures [6]-[10]. In the BPM FEM approach presented in [7] and the modal analysis [8] the PMLs are of rectangular type which are not the most suitable for optical fiber problems with cylindrical shapes and also our approach uses the Padé approximation. In addition, a FEM formulation is presented in [9], were a combination of edge elements for the transverse field and nodal elements for the longitudinal field is used together with a geometric transformation to deal with open domains.
The refractive index of the SIHF is showed in Fig. 1 and the coupler constituted of this type of fibers with two and three cores is showed in Fig. 2a and Fig. 2b, respectively. Perfectly matched layers (PML) of cylindrical type are used to reduce the computational domain.
The introduction of the small air hole produces an ultra-flattened chromatic dispersion in each fiber when analyzed separately. The behavior of the resulting geometry has been optimized by a genetic algorithm in conjunction with an efficient vectorial finite element formulation with instruction of perfectly matched layer of cylindrical type. The possible explanation could be the compensation between the waveguide dispersion resulting from the geometry and taking into account the effects of the air hole with the material dispersion [11]. The proposed structure shows a simple geometry when compared to other projects, especially when compared with the ones based on photonic crystal fibers due to the difficult to design and construct these kinds of fibers. In the coupler under consideration, the power can be split across two or three cores and forwarded to different locations in a small coupling distance by using the optimized fibers by genetic algorithms with the introduction of a small air hole at the center of the fibers. In the next section, the details of the coupler design are presented. The finite element formulation with cylindrical perfectly matched layers (CPML) is briefly explained in section 3. Numerical results including the sensitivity to geometrical parameters are presented in the section 4, and finally the main conclusions are given at the end.
II. COUPLER DESIGN
A. SIHF Design
The refractive index profile the SIHF used to construct the coupler is shown in Fig. 1(a) and the normalized propagation constants of the two first modes of this fiber with parameters 1 = 1.05214, 2
= 1.0, r1 = 0.2 m and r2 =1.8 m optimized through GA [2] are shown in Fig. 1(b). In this work, we
consider the coupler constituted by two and three cores as shown in Fig. 2a and Fig. 2b, respectively, separated by distance ℓ were nc and ns represent refractive index of the cores and the cladding,
respectively. Here nc is the refractive index of the pure silica and ns is the refractive index of silica
doped with germanium.
The refractive index profile n(r,) of an optical fiber can be written as n(r,) = (r) n(), where
n() is the wavelength refractive index and (r) is the normalized refractive index profile, which is assumed to be a function of the radial coordinate only. In this way, the refractive index of the step index holey fiber has been normalized in relation to the refractive index of pure silica and silica doped with germanium.
In addition, r1 and r2 represent the radius of the air and solid core, respectively. In this paper n = 1.0
at the air core center of the fiber and (r) for the normalized factor of the second ring core. The Sellmeier coefficients for pure silica and silica doped with germanium n() are taken into account. For
the specification of a wavelength dependent refractive index of an optical material, it is common to use a so-called Sellmeier equation [12] given by,
2 1 1 2 2 2 1 m i i i A n (1)In this work the value of the normalized refractive index of the core, which is doped with Ge, is
(r) = 1.05214 . This value was optimized by the genetic algorithm and used in all. By using the
optimized value of η(r) the effective refractive indexes (fundamental mode) were computed. They are shown in figure 1b which shows acceptable values for practical implementation.
(a) (b)
Fig. 1. (a) Refractive index profile of the SIHF and (b) normalized propagation constants of the two first modes of a fiber with parameters 1 = 1.05214, 2 = 1.0, r1 = 0.2 m and r2 =1.8 m.
The Fig. 2a and Fig. 2b shows the transversal section of the coupler constituted from two and three cores of a SIHF proposed in this work. The cores are indicated by numbers 1, 2 and 3, and its positions have been selected to keep the symmetry of the structure. The distance between the cores 1,
1.2 1.4 1.6 1.8 2.0 2.2 1.42 1.44 1.46 1.48 1.50 1.52 neff wavelength [m] fundamental mode second order mode silica
2 and 3 is constant along of the propagation direction. In the firstly analysis, the signal is launched in the core 1 and the coupling is verified in the core 2. Next, we consider the signal launched in core 1 and the coupling is verified in the cores 1 and 2. The size of air hole localized at the center of each core may be obtained and optimized in function of the horizontal coupling coefficient through GA. For all numerical calculations, the material that constitute the cores is the silica doped with germanium with refractive index nc. The cores are immerged in a cladding with refractive index
constituted by pure silica with refractive index ns.
(a) (b)
Fig. 2. Coupler constituted by (a) SIHF with two cores and (b) SIHF with three cores.
B. Genetic Algorithm
For the SIHF shown in Fig. 1, the optimized parameters were 1, 2, r1 and r2, where 1 and 2 are
the normalized refractive index of the core and cladding, respectively. The parameters of the structure in question belong to the set of real numbers, assuming, easily, fractional parts. Because of this characteristic, a binary encoding would increase the number of bits needed to represent the whole chromosome, and require a greater effort in the ongoing task of encoding, decoding and correction of values. Thus, it was decided to adopt values coding preserving the type of optimized variables. In the
GA was used ranking by selection and two points crossover. The population was composed by 100
individuals. The mutation tax was 1%, the elitism tax was 75% and generations’ number was 50 and crossover tax was 80%. In order to start the genetic algorithm life cycle, the population vector with new individuals (chromosomes) is adjusted. Before generating the new chromosomes, the computational code needs to define all the possible values for each gene. The new population of chromosome is evaluated as shown in Fig. 3.
nc ℓ Core 1 nc ns z x y PML r2 Core 2 2r1 nc PML ℓ 2r1 ℓ r2 nc ns Core 1 Core 2 Core 3 y x z
Fig. 3. Stages of the evaluation of chromosome population.
In this work we analyze the SIHF for application in the analysis of the coupler constitutes from two and three these models of fibers. To evaluate each optical fiber configuration, we use an extern computational code to find the eigenvalues. This code is executed just before the fitness function and the output is a data file that is later processed to obtain the chromatic dispersion as function of the wavelength. As any other optimization method, genetic algorithms have their performance related not only to their internal functions but also to the computer architecture used, since the high processing power, the great number of individuals evaluated in the same amount of time. In this study we used a Windows 7 operational system compiled to take advantage of the 64-bit instruction set architecture present on Pentium Core i7 with some specifications such as 6 GB of RAM. The fitness function is a simple code that reads the dispersion file and creates a matrixmx2. The first matrix column represents
the wavelength and the second column is the chromatic dispersion. Next, the algorithm creates two variables to store the maximum (max) and minimum (min) values for the chromatic dispersion. In this work the focus of the genetic algorithm is used to obtain flattened (next zero) chromatic dispersion. The flattened dispersion, is calculated the range position to obtain the dispersion close of the zero. After the chromatic dispersion is obtained, we use the fiber model for which this dispersion was calculated to construct the couplers showed in Fig. 2.
III. FINITE ELEMENT FORMULATION
A. Finite Element Formulation
Modeling Component Fitness Function Modeling Software Data Converter Coupling Distance Calculation Start End GA VBPM
In summary, the formulation is obtained based on the Helmholtz’s vectorial equation in two dimensions and considering cylindrical perfectly matched layer (CPML) to avoid undesired reflections. Thus, it may write:
0 2 0 k H k H (2) where L
k 1 , represents the relative permissiveness tensor and L represents tensor relative
parameters to the cylindrical PML or mean with virtual losses relative parameters. Considering the dielectric mean with transversal anisotropy and defining uˆx, uˆy and uˆz associated to the directions x, y
and z, respectively, may be written as xxuˆxuˆxxyuˆxuˆyyxuˆyuˆxyyuˆyuˆyzzuˆzuˆz
and z z zz y y yy x y yx y x xy x x xxuu L uu L uu L uu L uu L L ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
. After some algebraic simulations and applying the finite element method for the traversal variation, there is the following differential equation:
{ } 2
{ } (
2
){ } {0} 2 2 T T T h M K z h M z h M (3)where {hT} represents a column vector which contains the unknown components hxj and hyj, {0} is the
null column vector, but [M] and [K] are called global matrices defined in [6]. Applying the Padé approximation (1,1) in equation (3), it could be rewritten as follows:
~
T
0 T h K dz h d M (4) where
( ) 4 1 ~ 2 2 K M M M . Finally, the finite difference method is applied for the variable z,
turning the problem into an algebraic equation system of the form:
M z zK z
hT z z
M z zK z
hT z ) ( ) 1 ( ) ( ~ ( ) ) ( ) ( ~ ( (5)where z is the propagation step in direction of z and (0 1) is inserted to control the method stability. Stability studies confirm that this method is unconditionally stable to 0.5 1. For = 0.5 in (5), it is already known as the Crank Nicholson’s algorithm. In addition, material dispersion is directly introduced in our calculations trough the expression: 𝐷 = −(𝜆 𝑐⁄ )(𝑑2𝑛
𝑒𝑓𝑓(𝜆) 𝑑𝜆⁄ 2) where
neff is the effective refractive index is the wavelength and c is the free-space speed of light.
B.
Perfectly Matched Layer of Cylindrically TypeThe perfectly matched layers are used as mathematical tool to truncate the computational domain. In the formulation presented in the previous section, the parameters of the PML are directly
the scheme used in the formulation is implemented in Cartesian coordinates and the tensor elements
for the cylindrical PML, are represented in terms of cylindrical coordinates (ρ, φ, z ). Thus, the
L
tensor is given by: z z z z z S S S S S S s S S S S S S S S S S S S S S s S L 0 0 0 cos sin sin cos 0 sin cos sin cos 2 2 2 2 (5)were Sφ, Sρ and Sz are the stretching variables in the cylindrical coordinate system. Since the waves are
assumed to propagate along the z direction, the parameters Sφ and Sz are set to unity in all regions of
the structure shown in Fig. 1 and Sρ takes the form:
PML PML PML PML d i S , 1 , 1 ) ( 2 (6)
where is the attenuation parameter. The thickness of the PML (dPML) is chosen so that the imaginary
part of the effective index converges to a stable value [10]. In all calculation PML is the cladding
radius and represents the total radius of the structure that constitutes the coupler. IV. RESULTS
In order to obtain a fiber with optimum dispersion characteristics, we considered several fiber configurations and the best result has been obtained considering a step index holey fiber with the following parameters 1 = 1.05214, 2 = 1.0, r1 = 0.2 m and r2 =1.8 m. From these parameters, we
started to analyze the coupling characteristics for the best case in which the dispersion obtained was flat and varies between -2 and +2 ps/(km.nm) from 1.2 m to 2.2 m as can be seen in Fig. 4 (squared black line).
Fig. 4. Chromatic dispersion as function of the core refractive index.
Next, we analyze the effects of the wavelength variation on the max transference of energy between the two cores shown in Fig. 2. The transference of energy can be controlled varying the distance between the cores or varying the wavelength. In this work we consider two values for distance between the cores, ℓ = 0.5 m and ℓ = 1.0 m, and we evaluated the results considering the following values for the wavelength, = 1.2 m, = 1.55 m and = 2.2 m.
The total CPU time required to obtain each dispersion curve with 60 points was less than 1.2 minutes running in a Pentium Core” i7 6 GB of RAM. In the simulations considering the structure with two cores, the signal was launched in the left core with a beam corresponding to the mode
E
11x, whose the effective index (neff) was calculated through the relation/ K0, obtained separately for oneof the fibers through the modal analysis [4]. In addition the coupling length L obtained using the full-vectorial finite- element beam propagation method (VBPM) analysis for two applications with SIHF as couplers is in agreement with the value obtained through the relation LB/(eff1eff2) using the
well know modal analysis [4], where LB is the beating length, and eff1 and eff2 corresponding the
propagation constant of the super modes symmetrical and anti-symmetrical of lower order. The comparison with the results obtained from modal analysis to confirm the accuracy of the proposed method, since that modal analysis technique is recognized accurate for these types of application.
In order, to obtain an ultra-flat dispersion, the SIHF was optimized using one evolutionary technique based in genetic algorithms (GA) in combination with the finite element method [2]. In addition, the results obtained in this paper for structure showed in Fig. 2 were obtained using a circular computational window with radius about 28.8 m divided in nearly 10,000 linear elements (preliminary calculations showed that the use of finer meshes does not improve the accuracy of the numerical results) and propagation step z = 0.1 m.
The Fig. 5 shows the normalized power along the z-direction for coupler shown in Fig. 2, where
1.2 1.4 1.6 1.8 2.0 2.2 -20 -10 0 10 20 30 40 50 c h ro ma ti c d is p e rs io n [ p s /n m.k m] wavelength [m]
conventional step index fiber step index holey fiber 1=1.06214 step index holey fiber 1=1.05214
core 1 (simple line) and core 2 (line with symbol), considering ℓ = 0.5 m, 1 = 1.55 and 2 = 1.2 m.
Fig. 5. Maximum distance of power transfer for SIHF with two cores for considering ℓ = 0.5 m, 1 = 1.55 and 2 = 1.2 m.
The Fig. 6 shows the normalized power along the z-direction for coupler 2, where core 1 (simple line) and core 2 (line with symbol), considering ℓ = 1.0 m, 1 = 1.55 and 2 = 1.2 m.
Fig. 6. Coupler 2.
Figure 7 shows the normalized power along the z-direction for coupler 3 considering ℓ= 1.0 m and = 2.2 m. 0 50 100 150 200 250 0,0 0,2 0,4 0,6 0,8 1,0 2=1.20 m 1=1.55 m
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Core 1 Core 2 0 100 200 300 400 500 0,0 0,2 0,4 0,6 0,8 1,0 2=1.20 m 1=1.55 mN
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Core 1 Core 2Fig. 7. Coupler 3.
In the third application, we consider the structure shown in Fig. 2b. Here, the VBPM was applied to calculate the coupling distance for SIHF with three cores with air hole in the center at the core (coupler 4). In this case the structure was stimulated with a beam launched in the central core (core 1) and the maximum power transfer occurred for a distance around 500 µm. In addition, the multicore optical fibers play a significant role in dividing/combining of the power in optical fiber networks [3], [11]. On these fiber models, a single power may be divided through the cores in the structure and forwarded to different areas for additional purposes. In this case, the approach suggested establishing the mutual contact between the three identical cores is the fusion process [10]. This fusion process results in a tapered region of fused optical material, where the power transfer among fibers occurs due to their proximity [3],
The results for the coupler4 is showed in Fig. 8 and represents the normalized power along the z-direction for coupler constituted from SIHF with three cores, where the black curve with empty circles corresponds to the power variation of the central core or Core 1, the red curve with full circles corresponds to the power coupled to left horizontal core or Core 2 and the blue curve with semi-full squares circles corresponds to the power coupled to right horizontal core or Core 3, considering ℓ = 1.0 m and wavelength = 1.55, in the propagation direction. In this simulation, the radius of the circular computational window used was 28.8 m divided in nearly 15,000 linear elements and the propagation step used was z = 0.1 m.
It may be noticed from the numerical results that the powers in the cores 2-3 gradually increase, while the whole power launched in the central core is entirely transferred to the adjacent cores in propagation distance around 380 μm.
0 100 200 300 400 500 0,0 0,2 0,4 0,6 0,8 1,0
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Core 1 Core 2Fig.8. Coupler 4.
Next, in the coupler 5 (Fig. 9) it is shown the results for the normalized power along the z-direction for coupler constituted from SIHF with three cores considering ℓ = 1.0 m and wavelength = 1.20
m. Here the black curve with empty circles corresponds to the power variation of the central core or Core 1, the red curve with full circles corresponds to the power coupled to left horizontal core or Core 2 and the blue curve with semi-full squares circles corresponds to the power coupled to right horizontal core or Core 3. The mesh parameters used in this simulation were the same used for obtaining the coupler 4. In this case, the powers in the cores 2-3 gradually increase too, while the whole power launched in the central core is entirely transferred to the adjacent cores in propagation distance around 1000 μm.
Fig.9. Coupler 5.
Preliminary results based on sensitivity analysis in the distance between the cores (ℓ) and wavelength of the SIHF shown that small variations in this parameters, while the others ones remain
0 100 200 300 400 500 0,0 0,2 0,4 0,6 0,8 1,0
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Core 1 Core 2 Core 3 0 200 400 600 800 1000 1200 0,0 0,2 0,4 0,6 0,8 1,0N
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Core 1 Core 2 Core 3constants, can cause significant variations in the coupling distance. In addition, the losses presents in these types of couplers are neglected due of the signal propagation to occur by total internal reflection. These couplers are very sensitive to changes in the wavelength, so they are widely used as wavelength division multiplexing (WDM) for application in telecommunication systems that combine multiple input channels with different wavelengths, or to separate channels.
V. CONCLUSIONS
In this paper, the vectorial beam propagation method (VBPM) in conjunction with cylindrical perfectly matched layer (CPML) and GA have been successfully applied to the analysis of the coupler constituted from two and three cores immersed in a SIHF. The present formulation permits the insertion of cylindrical PML conditions. Sellmeier’s equations were applied to determine the refractive indexes of the pure silica and silica doped with germanium. The analyzed structures exhibit a simple geometry when compared with some structures based in photonics crystal fibers used as couplers. The parameters 1, 2, r1 and r2 were optimized through GA. The results obtained with two
cores show that the power launched in the core 1 for second application in entirely transferred to adjacent core 2. When we consider the coupler composed by three cores, the power launched in the central core has been equally divided between two neighbors and equidistant cores with a coupling ration around of 50% as expected. The idea of doping the core with Ge, the optimization of the SIHF by using GA and the application of this fiber model to analysis of a directional coupler are the main novelties of this work, resulting in an ultra-flattened chromatic dispersion fiber and it also can be used as a coupler device of rapid energy transfer along the propagation distance with neglected losses.
ACKNOWLEDGMENT
The authors would like acknowledging the UFRN, UFERSA and FINEP for the physics support, and INCT Fotonicom and CNPq for financial support.
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