Photonic Crystal Fiber With an Ultrahigh ... - OSA Publishing

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 31, NO. 2, JANUARY 15, 2013

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Photonic Crystal Fiber With an Ultrahigh Birefringence and Flattened Dispersion by Using Genetic Algorithms I. Abdelaziz, F. AbdelMalek, S. Haxha, Senior Member, IEEE, H. Ademgil, and H. Bouchriha

Abstract—We developed a high-throughput technique to design photonic crystal fiber (PCF) structures with desired properties and functionalities. By using a genetic algorithm, a high birefringence and an ultra-flattened chromatic dispersion over a large wavelength range are achieved. It is shown that a low confinement loss can be obtained while the birefringence remains the same. The numerical results show that the presented PCF structure can be successfully employed as maintaining polarization devices working in a large zerochromatic dispersion region. Index Terms—Chromatic dispersion, birefringence, finite element methods, genetic algorithms, photonic crystal fibers.

I. INTRODUCTION

P

HOTONIC CRYSTAL FIBERs (PCFs), also called holey fibers or microstructured fibers, have attracted many research groups. PCFs consist of air holes running parallel to its axis over the whole length. The guidance of light through these fibers is governed by the type of these structures. Hollow core fibers generate guidance by the photonic band gap, this characteristic result in the ability to forbid propagation of light within a frequency range [1], [2]. However in case of the solid core fibers, the light guidance can be achieved by total internal reflection. PCFs with solid core offer more design possibilities by varying the number of air holes in the cladding and lattice constant, leading to many applications that cannot be achieved with traditional optical fibers. The tremendous progress in telecommunications, materials processing and health services are increasing the demand for either chemical or biochemical sensing of low index materials. Also, using PCFs in industrial level may lead to many benefits such as miniaturization, high degree of integration and remote sensing capabilities [3], [4]. In order to increase the ability of conventional optical fibers, several engineering approaches were introduced such as; performing tapers Manuscript received June 20, 2012; revised September 19, 2012; accepted October 22, 2012. Date of publication November 12, 2012; date of current version December 28, 2012. I. Abdelaziz, F. AbdelMalek, and H. Bouchriha are with the National Institute of Applied Sciences and Technology, BP 676 Cedex 1080, University of Carthage, Tunis, Tunisia. S. Haxha is with the School of Engineering and Digital Arts, The University of Kent, Canterbury, Kent, U.K.. H. Ademgil is with the Department of Electronics, European University of Lefke, Mersin 10, Turkey. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JLT.2012.2226866

in those fibers, whereas all these approaches make the geometry of the fibers more complex and easy to break down. Compared to conventional optical fibers, these microstructures exhibit a number of unusual properties, including single-mode operation at all wavelengths [5], optical nonlinearity, high birefringence, and anomalous dispersion in the visible and near-infrared regions [6], [7]. Moreover, the optical gain provided by PCFs is under intensive study [8]. Therefore, PCFs can find wide applications in many research areas, including telecommunications and medicine [4]. The proposed PCFs may not be simple to fabricate. However, the current progress in PCF technology has demonstrated that fabrication of even more complex PCF structures is possible. Unfortunately, possible errors during the fabrication process, as in any PCF structure, could effect the chromatic dispersion. Fabrication error possibilities, such as the variation of hole size and hole-to-hole spacing (between 0.5 m–2.5 m) in the dispersion flatness, has been demonstrated by F. Poletti et al. [9]. Our simulations results have indicated that, chromatic dispersion is rather sensitive when air hole size and the position of inner rings are changed. Similar behaviour has been demonstrated by Poletti et al. [9]. PCF dispersion and birefringence have also been investigated theoretically and experimentally by Suzuki et al. [10]. They [10] have demonstrated that PCFs with constant is easy to fabricate with a conventional multiple-capillary drawing method. In our proposed PCF design, is also m) than in the PCF structure constant and larger ( presented in [9]. Therefore, fabrication of the proposed PCFs is believed to be attainable with a high feasibility and is not beyond the realms of today’s available technology. Another alternative fabrication method is to use the sol-gel method which allows for independent adjustment of the hole size and spacing. The Sol-gel method [11] provides additional design flexibility that will be necessary for such PCF structures. PCFs have been used to realize a polarization-maintaining device [12], thus in order to achieve this goal, the modal birefringence should be large. Many designs have been proposed aiming in increasing the birefringence by introducing asymmetric core and keeping a periodic cladding to minimize the confinement losses [13]. M. Chen et al., proposed a modified core by considering four elliptical air holes surrounding a central rod [14]. Also, PCFs find their application in THz regime, in order to achieve this regime, the refractive index in core should be reduced compared to that of the cladding region. In [15] K. Nielsen et al., have introduced a sub-wavelength structure in the twin cores in order to design a directional coupler operating

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in the THz region. Recent designs of high birefringence PCFs have been proposed [16] among which a birefringence of about was achieved [16]. In this study, an ultrahigh birefringent PCF with five rings where the radius of air holes varies from one ring to another one, is proposed. The performances of the proposed PCF are determined by using a genetic code approach. The genetic code is introduced to design a PCF with an ultrahigh birefringence and zero chromatic dispersion over a large wavelength range. The genetic code has been used to optimize the structural parameters along with the topology of a photonic crystal [17]. In the presented structure, we employ the genetic code to optimize the diameters of the air holes and the pitch in order to achieve a high birefringence PCF. Optimization ensures a performance towards desired goals. The Genetic algorithm approach has flexibility, presents advantages and is able to optimize the PCF parameters. This approach is based on adopting the principles inspired from natural population genetics to structure solutions to the given problem. The procedure using the genetic code consists of creating a new set of artificial creatures in every generation. The genetic algorithm method uses a rich database of points simultaneously compared to other methods. This code is more accurate because the probability of finding a false data is minimal. The method requires the natural parameter set of optimization issue to be coded as a finite length string. In this study, a developped matlab code is used to extract and optimise the structural parameters of the PCF. In the optimization process these parameters have to be coded as a finite length string.

The selection phase consists of determining how many times the structure is selected to generate other ones. This selection is based on the performances of each structure where the error function is small. The probability to select the th structure is (2)

Once the parameters of the structure are introduced and coded, we calculate the modes of the proposed PCF for each set of structural parameters. We employ the finite element method (FEM). Using the FEM, the cross-section of the PCF and basically the area containing the air holes is divided into homogeneous subspaces. In the FEM the subspaces are considered as triangle element to fit the circular shapes where Maxwell’s equations are solved. As boundary conditions, we use an anisptropic perfectly matched layer (PML) [16]. From Maxwell’s equations, the curl equations are given as: (3) Where presents the electric field vector, is the wave number in the vacuum, is the refractive index of the domain, where it is set to 1.45, is the PML matrix, is an inverse matrix of and is the operating wavelength. The dimensions of the computational window in - and -direction are m and m, the width of the PML, is 2 m. B. Validation of the Genetic Algorithm

II. RESULTS AND DISCUSSIONS In this work, the genetic algorithm based computation is developed and employed. The desired performances such as zerochromatic dispersion on a large band and high birefringence are fixed. The main goal is to reach these values. A. Genetic Algorithm The Genetic algorithm technique has been exploited to investigate fundamental properties in PCFs and this is reported in many papers [18], [19]. We start by coding the parameters of the structure to be optimized; hence each population is treated as a set of structures called chromosomes. These structures contain large number of genes, which are identified as the structural parameters. The hole spacing and the ratio of diameters of air holes to : are stored in the same array. For each individual, we define an error function given as (1) Where represents the target value of the constraint of the genetic code, and is the calculated value at for the same constraint.

In order to validate the developed code, we consider the structure proposed by T. P. Wite et al., [1], which is composed of one ring layer of air holes with a period (pitch) m. The diameter of the air holes is m. The target parameter is the real part of the refractive index and the operating wavelength m. This value was calculated and reported in [1] as 1.4453952. As an initial step, we fix the value of population used in the algorithm during the calculation, which is taken as 50 individuals. In order to achieve a rapid convergence of the algorithm the population was large. We calculate the error function for various values of the number of generation (g). The results are reported in Table I, we notice the error is small when g is 10. When we have increased g to 20; the calculated error function is reduced rapidly to 1.08 . We kept increasing the number of generation to 25; here we notice that the error is about . We further increased g to 30 the error function is slightly decreased to compared to the previous value. When g reaches the maximum value which is 49, the error function was performed to be around ; it remains roughly constant. From Table I, we notice that the larger the number of generations, then the more accurate result. Also, the error function fluctuates around when g varies from 25 to 40. In order to obtain accurate results and save computing time, we choose the number of generations to be fixed to .

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TABLE I THE ERROR FUNCTION DEPENDENCE ON THE NUMBER OF GENERATIONS

Fig. 2. Variation of the chromatic dispersion as a function of wavelength for the optimized structure.

TABLE II EVOLUTION OF THE CHROMATIC DISPERSION WITH THE SPECTRAL BAND

Fig. 1. Five-rings PCF with PML region.

III. DESIGN

FMAS STRUCTURE WITH FLATTEN CHROMATIC DISPERSION

OF

The structure under study consists of five rings of air holes arranged around the core region. The proposed structure is illustrated in Fig. 1 which consists of five rings where the radius of air holes varies from one ring to another. (one is proposed). We optimize such structure to achieve a flattened chromatic dispersion around zero over a large wavelength band varying from 1.2 m to 1.6 m. In this regard, we consider air holes of different and varying diameters. Chao et al. [3] demonstrated that by adopting this procedure the chromatic dispersion can be controlled over a large wavelength band and even kept equal zero [4] whilst at the same time the confinement losses are low. Next, in order to implement the genetic algorithm, we focus on the chromatic dispersion through it we calculate the error functions. Also we choose air holes of large diameters in the last ring of the structure to keep the confinement loss small. After the generation of 25 populations each of them is formed by 50 individuals, the desired properties such as the birefringence and chromatic dispersion are found. The optimized chromatic dispersion of the proposed structure is found to be around zero over large wavelength band with an error function equal to 0.39 ps/Km nm; its pitch is calculated to be 1.7825 m. The optimized air hole diameters in the five rings are respectively, 0.5381 m, 0.6157 m, 0.8384 m, 1.0158 m and 1.6769 m.

One may notice that these values vary from one ring to another, which is in agreement with our desired structure defined at the beginning. In order to be more confident with the result, in Fig. 2, we report the variation of chromatic dispersion with wavelength. This figure shows that chromatic dispersion varies from ps/Km.nm to ps/Km.nm, which fluctuates around zero. The fluctuation is maintained over a large wavelength band ranging from 1.335 m to 1.584 m. The dispersion is around zero over this large range. This result agrees with our desire aim. Also, the evolution of chromatic dispersion of the optimised structure over various spectral bands is illustrated on Table II. The confinement losses shown in Table II are around 0.014 for the three first spectral bands and that is about 0.03 for the last one, whilst the chromatic dispersion is around . The confinement losses of the fundamental mode is calculated and reported in Fig. 3. It can be noticed that the confinement varies linearly with wavelength over a large range from 1.2 m to 1.7 m. For small wavelengths the confinement loss is about however it increases to around when the wavelength is about 1.6 m. The confinement losses increase with wavelength, it varies from dB/Km for m to reach a value of dB/Km when m. It seems that the signal is not affected and keeps its form during propagation without distortion. Therefore, the proposed structure presents small loss compared to that reported in literature this means that the genetic algorithm is useful and can be implemented to optimize such a structure with specific requirements.

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Fig. 3. Confinement loss of the fundamental mode variation with wavelength. d, (a): Chromatic dispersion versus wavelength when d varies as d, and d, (b) Chromatic dispersion versus wavelength when d varies d, d, and d. around

In order to study the fabrication errors effects on the performances of the proposed structure, we vary the diameter of the air holes in the first ring. The result is reported in Fig. 3(a). This figure shows that the chromatic dispersion is almost zero when the wavelength is around 1.4 m for all fluctuations of the diameters of the air holes in the first ring. This means that at this wavelength fabrication errors do not affect severely the structure performance. Hence, the tolerance is about d. Also, the effect of the pitch d is thoroughly investigated and obtained the result is reported in Fig. 3(b). Fig. 3(b), shows that the fluctuation of the pitch affects strongly the chromatic dispersion regardless the wavelength region. IV. PCF WITH AN ULTRAHIGH BIREFRINGENCE In this section we focus on achieving a high birefringence over a large wavelength band varying from 1.3 m to 1.6 m. We define the birefringence as the difference between the refractive indices and . The proposed structure is illustrated in Fig. 4(a). which consists of five rings; each is formed by an alternative distribution of air holes of diameters and . Howand , ever, the air holes in the fifth ring have diameters where the diameter is larger than and . In order to increase the birefringence, we break the symmetry of the structure by shifting the air hole positions in the first ring by a distance equal to and remove two air holes of diameter from the first ring; this may allow the propagation in a specific direction. When we perform the calculation, one may notice that after

Fig. 4. (a) Five-rings PCF with PML region (b) Different rings of PCF.

twenty five generations, we obtain a pitch of m and set of diameters as m, m and m. The error function is calculated for those . The error function is small which values to be means that the diameters of the air holes will be fixed and used through the rest of the paper. For such a configuration, we calculate the birefringence as a function of the wavelength; the result is shown in Fig. 5. We notice that the birefringence increases linearly with wavelength and reaches a maximum value of about at m. The obtained birefringence value is large compared to that reported in several designs [12]. The proposed PCF is easy to fabricate compared to those introducing elliptical holes. For these structural parameters, the proposed PCF presents important properties that may improve significantly the performances of maintaining polarization devices. Also, on the same figure the evolution of the confinement losses is shown; one may notice that the maximum is less than 3 dB/Km. The effect of the fabrication tolerance on the birefringence of the proposed PCF is studied and reported in Fig. 5(a).

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Fig. 6. Variation of nonlinear coefficient as a function of wavelength.

Fig. 5. Variation of the birefringence (left curve) and confinement loss right curve) as a function the wavelength obtained through the optimized structure. (a) Variation of the birefringence with the wavelength when the diameters of air d, d, and d. (b) Bireholes in the first ring vary around d, d. fringence versus the wavelength when the pitch fluctuates

It shows that birefringence deviates from the optimum value as the fluctuation increases. Also, the birefringence is higher for positive fluctuations compared to those with negative fluctuations. The fluctuation of the pitch is investigated, the results are shown in Fig. 5(b). This figure shows that the birefringence is high for negative fluctuations, however presents larger diviations for positive fluctuations. The quality of the propagating signal can be affected by the nonlinear effects inside the core. In this regard, we investigate the nonlinear coefficient of the proposed design as a function of the wavelength. The nonlinear coefficient is defined as the ratio of the nonlinear refractive m W [19] to the effecindex coefficient ( tive area . We perform the calculations of the nonlinear coefficient and in Fig. 6 we report its variation with the wavelength. It is clear from this figure that at m, the nonlinear coefficient is small compared to that obtained for small wavelengths. Also, Fig. 6 shows that the nonlinear coefficient decreases linearly with increasing wavelength. The calculated value is about 21 Km at m, this means that the effective mode area is large compared to that obtained for small wavelengths this is a good evidence that the signal quality is not affected severely. Therefore the signal is able to propagate down the core without dispersion and keeping the same polarization. This result is crucial for developing ultra compact micro optoelectronic systems.

Fig. 7. Variation of the coupling efficiency as a function of wavelength for the optimized structure.

The coupling efficiency of the proposed PCF is investigated, the performed calcuations are shown in Fig. 7. One may notice that the coupling efficiency of the optimized PCF varies slightly with wavelength and it is slightly decreases as the wavelength increases. As the coupling efficiency plays a crucial role in propagation of signals and information processing, it would be better to consider smaller wavelength regimes. V. CONCLUSION A novel design of photonic crystal fibers with varying air hole diameters is proposed and analyzed. By using the genetic algorithm several key structural parameters such as the air hole diameter and the air hole separation distance are accurately optimized. A high birefringence and small chromatic dispersion are achieved through the optimization of the geometric parameters by the use of the genetic algorithm. REFERENCES [1] R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science, vol. 285, p. 1537, 1999. [2] R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science, vol. 285, p. 1537, 1999. [3] C.-Y. Chao, W. Fung, and L. J. Guo, “Polymer microring resonators for biochemical sensing applications,” IEEE J. Sel. Topics Quantum Electron., vol. 12, no. 1, p. 134, Jan. 2006. [4] V. Sh Afshar, S. C. Warren-Smith, and T. M. Monro, “Enhancement of fluoresce,” Opt. Express, vol. 15, no. 26, pp. 17891–17901, 2007.

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[5] B. Kuhlmey, R. C. McPhedran, and C. M. de Sterke, “Modal cutoff in microstructured optical fibers,” Opt. Lett., vol. 27, p. 1684, 2002. [6] H. Choi, C. Kee, K. Hong, J. Sung, S. Kim, D. Ko, J. Lee, J. Kim, and H. Y. Park, “Dispersion and birefringence of irregularly microstructured fiber with an elliptic core,” Appl. Opt., vol. 46, p. 8493, 2007. [7] S. F. Kaijage, Y. Namihira, F. Begum, N. H. Hai, S. M. A. Razzak, S. M. A. Kinjo, T. Miyagi, K. Nozaki, and S. Zou, “Highly nonlinear and polarization maintaining octagonal photonic crystal fiber in 1000 nm regions,” in Proc. OECC, 2009. [8] G. Mingyi, J. Chun, and H. Weisheng, “Dual-pump broadband fiber optical parametric amplifier with a three-section photonic crystal fiber scheme,” SPIE, Nanofabrication: Technol., Devices, Appl., vol. 5623, pp. 300–307, 2005. [9] F. Poletti, V. Finazzi, T. M. Monro, N. G. R. Broderick, V. Tse, and D. J. Richardson, “Inverse design and fabrication tolerances of ultra-flattened dispersion holey fibers,” Opt. Express, vol. 13, pp. 3728–3736, 2005. [10] K. Suzuki, H. Kubota, S. Kawanishi, M. Tanaka, and M. Fujita, “Optical properties of low loss polarization maintaining photonic crystal fibre,” Opt. Express, vol. 9, pp. 676–680, 2001. [11] R. T. Bise and D. J. Trevor, “Sol-gel derived microstructured fiber: Fabrication and characterization,” in Proc. OFC, Washington, DC, Mar. 2005, vol. 3, Optical Society of America. [12] A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriage, B. J. Mangan, T. A. Birks, and P. S. S. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett., vol. 25, pp. 1325–1327, 2000. [13] I. Abdelaziz, F. AbdelMalek, H. Ademgil, S. Haxha, T. Gorman, and H. Bouchriha, “Enhanced effective area photonic crystal fiber with novel air hole design,” J. Lightw. Technol., vol. 28, no. 19, p. 2810, 2010. [14] M. Chen, S. G. Yang, F. F. Yin, H. W. Chen, and S. Z. Xie, “Design of a new type high birefringence photonic crystal fiber,” Optoelectron. Lett., vol. 4, pp. 19–22, 2006. [15] K. Nielsen, H. K. Rasmussen, P. U. Jepsen, and O. Bang, “Broadband terahertz fiber directional coupler,” Opt. Lett., vol. 35, p. 2879, 2010. [16] J.-M. Hsu and D.-L. Ye, “Polarization-maintaining photonic crystal fiber with ultra-high modal birefringence, progress,” in Proc. Electromagn. Research Symp., Suzhou, China, Sept. 12–16, 2011.

[17] P. I. Borel et al., “Topology optimization and fabrication of photonic crystal structures,” Opt. Express, vol. 12, pp. 1996–2001, 2004. [18] R. R. Musin and A. M. Zheltikov, “Designing dispersion-compensating photonic- crystal fibers using a genetic algorithm,” Opt. Commun., vol. 281, pp. 567–572, 2008. [19] A. Huttunen and P. Töarmä, “Effect of wavelength dependence of nonlinearity, gain, and dispersion in photonic crystal fiber amplifiers,” Opt. Express, vol. 13, p. 4286, 2005.

S. Haxha (SM’06) received the M.Sc. and Ph.D. degrees from City University, London, U.K., in 2000 and 2004, respectively. His Ph.D. degree was directly funded by QinetiQ, Malvern, U.K. In 2003, From February 2004 he was a member of the Broadband and Wireless Communications group in the School of Engineering and Digital Arts, Kent University, Canterbury, U.K. as a lecturer. He has published research papers in major international and national media such as IEEE, Optics Express, Applied Optics (OSA), IET and Optics Communications. His research expertise is focused on designing and optimizing optical and microwave devices for applications in telecommunications and sensing systems. His research interests are in the areas of photonic crystal devices, metamaterials, photonic crystal fibres, surface plasmon polaritons (biosensors), ultra-high-speed electro-optic modulators and Optical Code Division Multiple Access (OCDMA) systems. From December 2008 to December 2011, Dr. Haxha was a CEO of the largest telecommunication company in Kosova, PTK, consisting of mobile operator and fixed line business units. Dr. Haxha was awarded the SIM Postgraduate Award from The Worshipful Company of Scientific Instrument Makers in Cambridge, UK for his highly successful contribution in research. He has obtained several world class E.M.B.A. and M.B.A. diplomas, and training in telecommunication, finances and management. He has been a key note speaker of numerous world class conferences related to telecommunication and management. He is listed in Who’s Who in the World.