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Numerical study of photonic crystal fiber with ultra-flattened chromatic dispersion in anomalous and normal dispersion regimes
Optik 126 (2015) 1307–1311
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Numerical study of photonic crystal fiber with ultra-flattened chromatic dispersion in anomalous and normal dispersion regimes A. Barrientos-García a,∗ , J.A. Andrade-Lucio a , Igor A. Sukhoivanov a , Igor Guryev b , J. Ruiz-Pinales a , R. Rojas-Laguna a a Departamento de Ingeniería Electrónica, División de Ingenierías, Campus Irapuato-Salamanca, Universidad de Guanajuato, Carretera Salamanca-Valle de Santiago km 3.5 + 1.8 km, Comunidad de Palo Blanco, Salamanca, Gto. 36885, Mexico b Departamento de Estudios Multidisciplinarios, División de Ingenierías, Campus Irapuato-Salamanca, Univerisdad de Guanajuato, Av. Universidad s/n, Col. Yacatitas, Yuriria, Gto. Cp. 38940, Mexico
a r t i c l e
i n f o
Article history: Received 25 February 2014 Accepted 1 April 2015 Keywords: Zero dispersion wavelength Fiber characterization Photonic crystal fibers
a b s t r a c t Here we propose a simple design for a photonic crystal fiber using the golden ratio parameter of pitch and four rings with hexagonal structure of air holes with solid core of silica. By varying the ratio of air hole diameters inside the structure we can control the range of flattened dispersion in anomalous or normal dispersion regime close to zero dispersion wavelength (ZDW). The wide range of wavelengths of 1.19–1.83 m with small losses for chromatic dispersion of −0.34 ± 0.66 ps/(nm-km) and 1.292–1.555 m with chromatic dispersion of 0 ± 0.06138 ps/(nm-km), can allow the possibility of use in different engineering areas, such as supercontinuum generation, pulse reshaping, and dense-WDM (DWDM) based optical communication systems. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Since the invention of the photonic crystal fiber (PCF) [1,2], many types and geometries of PCFs have been designed for working in several wavelength windows [3,4]. Other structures have been designed for working in a wide range of wavelengths in the area with zero chromatic dispersion wavelength (ZDW) [5–8]. The PCF with flattened dispersion in a wide range can be used in a wide field of engineering applications such as wide-band supercontinuum generation, dispersion compensation, ultra-short soliton pulse transmission, optical parametric amplification, wavelength-division multiplexing transmission, pulse reshaping or as a candidate for nonlinear optical processes [6–12]. This design of an Aureus-PCF (Au-PCF with the pitch value of 1.618 m) is particularly attractive because it contains a small number of rings, high sensitivity for controlling the chromatic dispersion by only changing the diameter of the third, fourth rings (or both at the same time) according to the application and type of dispersion needed (anomalous or normal dispersion), keeping an ultra-high flattened chromatic dispersion in a wide spectrum of wavelengths. In [13], the authors show numerically the possibility of raising an ultraflattened chromatic dispersion with low losses.
∗ Corresponding author. Tel.: +52 4646479940. E-mail address: [email protected] (A. Barrientos-García). http://dx.doi.org/10.1016/j.ijleo.2015.04.005 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
Other studies of PCFs using the golden ratio parameter in different ways than in this paper, which makes possible to obtain a wide variety of polarizations and waveguide dispersion behavior [14]. The PCF proposed here was analyzed using the finite element method (FEM) [15] and scattering boundary condition to determine the perfect match layers (PML) to solve the model, implemented in the commercial software COMSOL Multiphysics. We show the numerical analysis and different chromatic dispersion behavior that give us the possibility of realizing this design for ultra-flattened chromatic dispersion, and according to the parameters used here, thus it is possible to use the standard PCF fabrication [7,16] for this design of Au-PCF with d1 = 0.5 m, d2 = 0.6815 m and d3 ,. . ., dn = 1 m and only two sub-micron air holes rings in the structure of Au-PCF (18 air hole). In the literature, it can be found a larger number of applications designed to work in anomalous and normal dispersion close to ZDW. For instance, an ultra-flattened dispersion was proposed in [6], in which for the range of wavelengths from 1.23 m to 1.72 m, a chromatic dispersion of 0 ± 0.4 ps/(km-nm) was obtained. Other examples of ultra-flattened dispersion can be found in [7] where the authors reported a chromatic dispersion of 0 ± 0.6 ps/(nm-km) for wavelengths in the range 1.24–1.44 m. A PCF with ultra-flattened dispersion of 0.2 ± 0.2 ps/(km-nm), for 1.14–1.7 m was reported in [8]. Another PCF with flattened dispersion for a telecommunication window was designed in [3], the authors report different possibilities of obtaining normal, anomalous, and zero dispersion
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A. Barrientos-García et al. / Optik 126 (2015) 1307–1311
Fig. 1. Cross-section of an Au-PCF with four periods with diameters of air holes d1 , d2 , d3 , d4 and = 1.618 m.
curves (around 1.55 m and 0.8 m). The fabrication of this type of PCF is non-trivial because the required hole diameters are submicron and the existent are designed for this kind of fiber implying four periods with different hole diameters for ultra-flattened dispersion or including 11 periods with the same hole diameters [6,7] for a wide range of wavelengths and low losses. In this paper, we propose a simple Au-PCF with ultra-flattened dispersion in a wide range using a golden ratio parameter for pitch scaling to micron (1.618 m) and four periods with an hexagonal structure that surround the solid core of silica, and air hole diameters of d1 = 0.5 m, d2 = 0.675–0.7 m, and d3 = d4 = 0.85–1.0 m. The mode field diameter (MFD) presented in this Au-PCF shows small effective area with low confinement losses necessary to nonlinear optical processes and applications [13,17,18]. 2. Au-PCF design for ultra-flattened dispersion The PCF for ultra-flattened dispersion reported in the literature [6–8] consists of complicated structures with seven or more periods of rings, triangular lattices with pitch and air holes of diameters d. Here the Au-PCF presents four periods with three different air hole diameters and high sensitivity to control the chromatic dispersion curve by only varying the air hole diameters of the two last periods of the structure or varying only the second air hole diameter, obtaining a higher flattened dispersion in a wide range of wavelength spectrum over 640 nm. We show the cross section of the Au-PCF structure in Fig. 1. This structure is formed by four air hole rings with three different diameters d1 , d2 , d3 , and d4 where d4 could be equal to any of those diameters, for controlling the position of chromatic wavelength dispersion close to ZDW. In this way, we can get anomalous, normal or both types of chromatic dispersion close to ZDW according to the application and the range of wavelength needed. For a simple design we controlled these parameters of the group velocity dispersion (GVD) by only varying d3 and d4 and keeping d1 = 0.5 m and d2 = 0.675–0.7 m. We can obtain many different chromatic dispersion behaviors close to ZDW, by varying d3 and d4 , in the ranges from 0.85 to 1.0 m, with high sensitivity to change the ranges of wavelength and the chromatic dispersion in the ranges from x ± 1 ps/(nm-km) to x ± 0.022 ps/(nm-km) (x could take values from −2 to 2 ps/(nm-km)). The GVD was analyzed including the material dispersion, applying the Sellmeier’s equation directly in material properties (varying the permittivity constant ε according to the changes of wavelength for silica) of the model, and in this way, we used the exact Eq. (1) to obtain the
Fig. 2. Total flattened chromatic dispersion (TFCD) for an Au-PCF with four periods with diameters of air holes d1 , d2 , d3 , d4 and = 1.618 m. The red curve shows the TFCD for five periods with air hole diameters equal to d3 = d4 = d5 = 0.9 m. The black and blue curves show the TFCD for four rings with d3 = d4 = 0.9 m and d1 = d4 = 0.5 m. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
curves of total chromatic dispersion of an Au-PCF using the data of mode index (neff ) or constant of propagation (ˇ), calculated. D() = −
d2 neff cd2
,
(1)
where c is the speed of light in vacuum, neff is the mode index and is the operating wavelength. For the next results, we consider the lattice (pitch), d1 , and d2 , constant and varying at the same time d3 = d4 = 0.85, 0.875, 0.9 and 1.0 m, and only to show the sensitivity of this design, we show some curves where we vary only d4 = d1 = 0.5 m. In Fig. 2 we have three curves with different parameters for an Au-PCF, in the black curve we have four rings, and diameters d3 = d4 = 0.9 m, in this case we have positive dispersion with 0.4455 ± 0.349 ps/(nmkm) in a wide range of wavelengths (1225–1728 nm) and the red curve shows the same parameters including one more air hole ring with diameters equal to d3 and d4 , here we can see that the influence of one more rings does not improve the flattened dispersion or range of wavelengths where it presents a flattened dispersion, but improves the losses for technological applications [13]. The blue curve was obtained by setting the diameter four equal to diameter one (0.5 m), only to show the control we have by varying the diameter of the last ring and we can see how the dispersion curve moves down and presents an ultra-flattened dispersion in communication windows with normal dispersion equal to −0.07425 ± 0.07424 ps/(nm-km) from 1519 nm to 1670 nm and keeping only three different air hole rings in the Au-PCF. Here we show that by only varying the last ring we can have control of the total chromatic dispersion according to the type of dispersion close to ZDW required. In Fig. 3 we can see a high ultra-flattened chromatic dispersion in a wide range and the sensitivity and control we have by varying the two last diameters of rings of the Au-PCF, the solid black and black (square marker) curves show the chromatic dispersion for a four rings structure of an Au-PCF. For the black curve (square marker), we can see that the best result in dispersion can be realized in a wide range of wavelengths, and the variation in dispersion that we have obtained is 1.712 ± 0.022 ps/(nm-km) from 1345 nm to 1530 nm and in the solid black curve we can appreciate a little increase in
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Fig. 3. Total ultra-flattened chromatic dispersion (TUFCD) for an Au-PCF with four periods with diameters of air holes d1 , d2 , d3 , d4 and = 1.618 m. The solid black and black (square marker) curves show the TUFCD for four rings with d3 = d4 = 1.0 m and d1 = d4 = 0.5 m.
dispersion but we have a higher range of wavelengths and closer to ZWD (1.65 ± 0.04 ps/(nm-km) from 1330 nm to 1560 nm). In Fig. 4 we show the control to displace the curve by varying d3 and d4 to normal dispersion or almost center the curve dispersion at the ZDW point. In the blue curve we can see an almost centered curve at ZDW with 0.068 ± 0.488 ps/(nm-km) from 1210 nm to 1780 nm and in the red curve we have a wider range of wavelengths in normal and anomalous dispersion and with a little difference in total dispersion (−0.34 ± 0.66 ps/(nm-km) from 1190 nm to 1830 nm). In the green curve we centered the ultra high-flattened dispersion in 0.00015 ± 0.06153 ps/(nm-km) from 1292 nm to 1555 nm, here we can see that when we move down the curve at the center of the point with ZDW, the total dispersion increases a little but the range of wavelengths where is present this dispersion increases a little too. This flattened-dispersion response
Fig. 5. Electric field profile for the Au-PCF with d1 = 0.5 m, d2 = 0.6815 m, d3 = d4 = 1.0 m and = 1.618 m. (a) Here we used a 1500 nm wavelength to calculate the EFP. (b) Mode propagation inside the core for Au-PCF. Here we show the fundamental mode calculated for 1500 nm and we can see that the higher concentration of field is transmitted inside the core of the Au-PCF.
of the PCF can be utilized in dense-WDM (DWDM) based optical communication systems [19,20]. Here we can conclude that the analysis done for this simple designed PCF achieves best flattened dispersion than reported in literature in this range of wavelength for green curve in different points centered close to ZDW varying the air hole diameter of the second, third or fourth period of Au-PCF and keeping the pitch equal to 1.618 m despite an ultra-flattened dispersion with little variation of chromatic dispersion close to ZDW. Such that if we decrease or increase in 0.018 microns or more the pitch, the ultra-flattened chromatic dispersion increases and moves down or up the ZDW point, for this reason it is important to keep constant the golden ratio parameter of pitch in m ( = 1.618 m). 3. Mode field diameter and electric field profile for Au-PCF
Fig. 4. Total flattened chromatic dispersion (TFCD) for an Au-PCF with four periods with diameters of air holes d1 , d2 , d3 , d4 and = 1.618 m. The green curve shows the TFCD for four periods with air hole diameters equal to d2 = 0.6815 m and d3 = d4 = 1.0 m with ultra high-flattened dispersion centered at the ZDW. The blue and red curves show the TFCD for four rings with d3 = d4 = 0.875 m and d3 = d4 = 0.85 m. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In Fig. 5(a) we show the electric field profile (EFP) calculated for 1500 nm, the propagation mode is distributed in the core of the Au-PCF, the energy of the fundamental mode is concentrated from the center until the first ring of the structure. We can appreciate the smooth distribution of the field with a little evanescence mode inside the core. The EFP has small variations for different diameters of d3 and d4 in the range proposed from 0.85 to 1.0 m and keeping d1 = 0.5 m and d2 = 0.675–0.7 m. From Fig. 5(b) we can see the confinement mode field guided by the core of the Au-PCF for 1500 nm according to the EFP shown in Fig. 5(a). From Fig. 5(b) we can appreciate the mode propagated inside the center of the core for an Au-PCF using the parameters in Fig. 4 for the green curve
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the black (point marker) curve we can see that the MFD distribution with d3 = d4 = 0.875 m varies from 2.8 to 3.8 m with respect to a wavelength in the range of 1200–1800 nm and in the red curve we can see a small variation in the MFD distribution with respect to the blue curve, for this reason we can say that if we keep the first two diameters equal to d1 = 0.5 m and d2 = 0.675–0.7 m and we vary only the two last diameters in the 3rd and 4th periods in the ranges proposed (0.85–1.0 m) to obtain a ultra-flattened dispersion and control the position of the dispersion behavior and the range of wavelength, the MFD is negligible in this range of variation which agrees with Fig. 6. We can appreciate in Fig. 5(b) that the electric field mode is guided inside the core of the Au-PCF because we have a solid core of approximately 2.74 m of diameter according to the pitch proposed for this structure inspired in the relation of nature presented in some leaves, animals and human beings and scaled in m for an Au-PCF structure of pitch. The small values of the MFD and effective mode area for PCFs indicate that these fibers have numerous potential applications in nonlinear optical devices and high performance optical coupling [17,25]. 4. Conclusions
Fig. 6. Mode field diameter and effective area for an Au-PCF. (a) Here we show the MFD for different air hole diameters in the last two periods of an Au-PCF. The MFD is 2.74 m approximately for an Au-PCF, and the variation in the range of wavelength for flattened dispersion is 2.8 m to 3.8 m. (b) Effective mode area agrees with the MFD.
We studied and proposed a simple structure for a PCF with an ultra-flattened dispersion of four rings and golden ratio parameter (in m) of pitch. Here we demonstrated numerically a high sensitivity to control some parameters by only varying the last two periods of the Au-PCF keeping a high-flattened dispersion in a wide range of wavelengths. We showed numerically that it is possible to obtain ultra-flattened dispersion of 0 ± 0.06168 ps/(nm-km) from 1292 nm to 1555 nm. In addition, we analyzed the MFD and showed the electric field distribution for an Au-PCF and we concluded that the Au-PCF for high ultra-flattened dispersion presented a fundamental mode propagation inside the core in a wide range with good confinement which helped us obtain some nonlinear optical processes with a short length Au-PCF. The high sensitivity obtained by varying the two last diameters of an air hole ring makes possible to obtain normal or anomalous dispersion in different ranges of wavelength close to ZWD in the second or third communication windows. Acknowledgements
(d1 = 0.5 m, d2 = 0.6815 m, d3 = d4 = 1.0 m and = 1.618 m). In this case the structure of the fiber concentrates all the energy inside the core in a wide range of wavelengths and we can see that the mode does not present a large evanescence mode outside the center. A mode field diameter is shown in Fig. 6 for two different AuPCFs with four periods keeping the first two diameters of air holes equal to 0.5 m and 0.7 m, 0.6815 m. Here we used Eqs. (2) and (3), where the first one corresponds to a normalized frequency of the PCFs analogy with single index fiber and the second one is the Marcuse formula to evaluate the MFD using the modified Veff number and keeping the cut-off condition for all the range of wavelengths (Veff = 2.405) [21–24]. Veff =
2 a eff
n2co − n2FSM ,
w 1.619 2.879 = 0.65 + + , 6 3/2 aeff Veff Veff
(2) (3)
where aeff is the effective core radius, is the operating wavelength, nco is the core index (for PCFs with solid core we can take the refractive index of silica), nFSM is the effective index of the fundamental space-filling mode and w is the half MFD (called the effective modal spot size), where the background index is assumed to be 1.45. In
This work was supported in part by the National Council for the Science and technology (CONACYT) of Mexico, and Alejandro Barrientos García is grateful to CONACYT for a student research grant No. 329409/336847. References [1] J.C. Knight, Photonic crystal fibres, Nature 424 (2003) 847–851. [2] P.S.J. Russell, Photonic crystal fibers, Science 299 (2003) 358–362. [3] A. Ferrando, E. Silvestre, P. Andrés, J.J. Miret, M.V. Andrés, Designing the properties of dispersion-flattened photonic crystal fibers, Opt. Express 9 (2001) 687–697. [4] K. saitoh, M. Koshiba, Highly nonlinear dispersion-flattened photonic crystal fibers for supercontinuum generation in a telecommunication window, Opt. Express 12 (2004) 2027–2032. [5] L. Zhao-lun, H. Lan-tian, W. Wei, Tailoring nonlinearity and dispersion of photonic crystal fibers using hybrid cladding, Braz. J. Phys. 39 (2009) 50–54. [6] K. Saitoh, M. Koshiba, T. Hasegawa, E. Sasaoka, Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion, Opt. Express 11 (2003) 843–852. [7] W.H. Reeves, J.C. Knight, P.St.J. Russell, P.J. Roberts, Demonstration of ultraflattened dispersion in photonic crystal fibers, Opt. Express 10 (2002) 609–613. [8] K. Saitoh, N. Florous, M. Koshiba, Ultra-flattened chromatic dispersion controllability using a defected-core photonic crystal fiber with low confinement losses, Opt. Express 13 (2005) 8365–8371. [9] A. Hartung, A.M. Heidt, H. Bartelt, Design of all-normal dispersion microstructured optical fibers for pulse-preserving supercontinuum generation, Opt. Express 19 (2011) 7742–7749.
A. Barrientos-García et al. / Optik 126 (2015) 1307–1311 [10] K.P. Hansen, Dispersion flattened hybrid-core nonlinear photonic crystal fiber, Opt. Express 11 (2003) 1503–1509. [11] N. Vukovic, N.G.R. Broderick, F. Poletti, Parabolic pulse generation using tapered microstructured optical fibres, Adv. Nonlinear Opt. 2008 (2008) 480362. [12] X. Wang, T. Hamanaka, N. Wada, K. Kitayama, Dispersion-flattened-fiber based optical thresholder for multiple-access-interference suppression in OCDMA system, Opt. Express 13 (2005) 5499–5505. [13] G. Renversez, B. Kuhlmey, R. McPhedran, Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses, Opt. Lett. 28 (2003) 989–991. [14] A. Agrawal, N. Kejalakshmy, J. Chen, B.A. Rahman, K.T.V. Grattan, Golden spiral photonic crystal fiber: polarization and dispersion properties, Opt. Lett. 33 (2008) 2716–2718. [15] F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, D. Felbacq, Photonic Crystal Fibres, Imperial College Press, London, UK, 2005. [16] J.C. knight, T.A. Birks, P.St.J. Russell, D.M. Atkin, All-silica single-mode optical fiber with photonic crystal cladding, Opt. Lett. 21 (1996) 1547–1549. [17] K. Miyagi, Y. Namihira, S.M.A. Razzak, S.F. Kaijage, F. Begum, Measurements of mode field diameter and effective area of photonic crystal fibers by far-field scanning technique, Opt. Rev. 17 (2010) 388–392.
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[18] M. Zhang, F. Zhang, Z. Zhang, X. Chen, Dispersion-ultra-flattened square-lattice photonic crystal fiber with small effective mode area and low confinement loss, Optik 125 (2013) 1610–1614. [19] R.K. Sinha, S.K. Varshney, Dispersion properties of photonic crystal fibers, Microw. Opt. Technol. Lett. 37 (2003) 129–132. [20] A. Ferrando, E. Silvestre, J.J. Miret, P. Andrés, Nearly zero ultraflattened dispersion in photonic crystal fibers, Opt. Lett. 37 (2003) 129–132. [21] K. Saitoh, M. Koshiba, Numerical modeling of photonic crystal fibers, J. Lightw. Technol. 23 (2005) 3580–3590. [22] M. Koshiba, K. Saitoh, Applicability of classical optical fiber theories to holey fibers, Opt. Lett. 29 (2004) 11739–11741. [23] D. Marcuse, Loss analysis of single-mode fiber splices, Bell Syst. Tech. J. 56 (1977) 703–718. [24] N.A. Mortensen, J.R. Folkenberg, M.D. Nielsen, K.P. Hansen, Modal cutoff and the V-parameter in photonic crystal fibers, Opt. Lett. 28 (2003) 1879–1881. [25] J. Liang, M. Yun, W. Kong, X. Sun, W. Zhang, S. Xi, Highly birefringent photonic crystal fibers with flattened dispersion and low effective mode area, Optik 122 (2011) 2151–2154.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Numerical study of photonic crystal fiber with ultra-flattened chromatic dispersion in anomalous and normal dispersion regimes A. Barrientos-García a,∗ , J.A. Andrade-Lucio a , Igor A. Sukhoivanov a , Igor Guryev b , J. Ruiz-Pinales a , R. Rojas-Laguna a a Departamento de Ingeniería Electrónica, División de Ingenierías, Campus Irapuato-Salamanca, Universidad de Guanajuato, Carretera Salamanca-Valle de Santiago km 3.5 + 1.8 km, Comunidad de Palo Blanco, Salamanca, Gto. 36885, Mexico b Departamento de Estudios Multidisciplinarios, División de Ingenierías, Campus Irapuato-Salamanca, Univerisdad de Guanajuato, Av. Universidad s/n, Col. Yacatitas, Yuriria, Gto. Cp. 38940, Mexico
a r t i c l e
i n f o
Article history: Received 25 February 2014 Accepted 1 April 2015 Keywords: Zero dispersion wavelength Fiber characterization Photonic crystal fibers
a b s t r a c t Here we propose a simple design for a photonic crystal fiber using the golden ratio parameter of pitch and four rings with hexagonal structure of air holes with solid core of silica. By varying the ratio of air hole diameters inside the structure we can control the range of flattened dispersion in anomalous or normal dispersion regime close to zero dispersion wavelength (ZDW). The wide range of wavelengths of 1.19–1.83 m with small losses for chromatic dispersion of −0.34 ± 0.66 ps/(nm-km) and 1.292–1.555 m with chromatic dispersion of 0 ± 0.06138 ps/(nm-km), can allow the possibility of use in different engineering areas, such as supercontinuum generation, pulse reshaping, and dense-WDM (DWDM) based optical communication systems. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Since the invention of the photonic crystal fiber (PCF) [1,2], many types and geometries of PCFs have been designed for working in several wavelength windows [3,4]. Other structures have been designed for working in a wide range of wavelengths in the area with zero chromatic dispersion wavelength (ZDW) [5–8]. The PCF with flattened dispersion in a wide range can be used in a wide field of engineering applications such as wide-band supercontinuum generation, dispersion compensation, ultra-short soliton pulse transmission, optical parametric amplification, wavelength-division multiplexing transmission, pulse reshaping or as a candidate for nonlinear optical processes [6–12]. This design of an Aureus-PCF (Au-PCF with the pitch value of 1.618 m) is particularly attractive because it contains a small number of rings, high sensitivity for controlling the chromatic dispersion by only changing the diameter of the third, fourth rings (or both at the same time) according to the application and type of dispersion needed (anomalous or normal dispersion), keeping an ultra-high flattened chromatic dispersion in a wide spectrum of wavelengths. In [13], the authors show numerically the possibility of raising an ultraflattened chromatic dispersion with low losses.
∗ Corresponding author. Tel.: +52 4646479940. E-mail address: [email protected] (A. Barrientos-García). http://dx.doi.org/10.1016/j.ijleo.2015.04.005 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
Other studies of PCFs using the golden ratio parameter in different ways than in this paper, which makes possible to obtain a wide variety of polarizations and waveguide dispersion behavior [14]. The PCF proposed here was analyzed using the finite element method (FEM) [15] and scattering boundary condition to determine the perfect match layers (PML) to solve the model, implemented in the commercial software COMSOL Multiphysics. We show the numerical analysis and different chromatic dispersion behavior that give us the possibility of realizing this design for ultra-flattened chromatic dispersion, and according to the parameters used here, thus it is possible to use the standard PCF fabrication [7,16] for this design of Au-PCF with d1 = 0.5 m, d2 = 0.6815 m and d3 ,. . ., dn = 1 m and only two sub-micron air holes rings in the structure of Au-PCF (18 air hole). In the literature, it can be found a larger number of applications designed to work in anomalous and normal dispersion close to ZDW. For instance, an ultra-flattened dispersion was proposed in [6], in which for the range of wavelengths from 1.23 m to 1.72 m, a chromatic dispersion of 0 ± 0.4 ps/(km-nm) was obtained. Other examples of ultra-flattened dispersion can be found in [7] where the authors reported a chromatic dispersion of 0 ± 0.6 ps/(nm-km) for wavelengths in the range 1.24–1.44 m. A PCF with ultra-flattened dispersion of 0.2 ± 0.2 ps/(km-nm), for 1.14–1.7 m was reported in [8]. Another PCF with flattened dispersion for a telecommunication window was designed in [3], the authors report different possibilities of obtaining normal, anomalous, and zero dispersion
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Fig. 1. Cross-section of an Au-PCF with four periods with diameters of air holes d1 , d2 , d3 , d4 and = 1.618 m.
curves (around 1.55 m and 0.8 m). The fabrication of this type of PCF is non-trivial because the required hole diameters are submicron and the existent are designed for this kind of fiber implying four periods with different hole diameters for ultra-flattened dispersion or including 11 periods with the same hole diameters [6,7] for a wide range of wavelengths and low losses. In this paper, we propose a simple Au-PCF with ultra-flattened dispersion in a wide range using a golden ratio parameter for pitch scaling to micron (1.618 m) and four periods with an hexagonal structure that surround the solid core of silica, and air hole diameters of d1 = 0.5 m, d2 = 0.675–0.7 m, and d3 = d4 = 0.85–1.0 m. The mode field diameter (MFD) presented in this Au-PCF shows small effective area with low confinement losses necessary to nonlinear optical processes and applications [13,17,18]. 2. Au-PCF design for ultra-flattened dispersion The PCF for ultra-flattened dispersion reported in the literature [6–8] consists of complicated structures with seven or more periods of rings, triangular lattices with pitch and air holes of diameters d. Here the Au-PCF presents four periods with three different air hole diameters and high sensitivity to control the chromatic dispersion curve by only varying the air hole diameters of the two last periods of the structure or varying only the second air hole diameter, obtaining a higher flattened dispersion in a wide range of wavelength spectrum over 640 nm. We show the cross section of the Au-PCF structure in Fig. 1. This structure is formed by four air hole rings with three different diameters d1 , d2 , d3 , and d4 where d4 could be equal to any of those diameters, for controlling the position of chromatic wavelength dispersion close to ZDW. In this way, we can get anomalous, normal or both types of chromatic dispersion close to ZDW according to the application and the range of wavelength needed. For a simple design we controlled these parameters of the group velocity dispersion (GVD) by only varying d3 and d4 and keeping d1 = 0.5 m and d2 = 0.675–0.7 m. We can obtain many different chromatic dispersion behaviors close to ZDW, by varying d3 and d4 , in the ranges from 0.85 to 1.0 m, with high sensitivity to change the ranges of wavelength and the chromatic dispersion in the ranges from x ± 1 ps/(nm-km) to x ± 0.022 ps/(nm-km) (x could take values from −2 to 2 ps/(nm-km)). The GVD was analyzed including the material dispersion, applying the Sellmeier’s equation directly in material properties (varying the permittivity constant ε according to the changes of wavelength for silica) of the model, and in this way, we used the exact Eq. (1) to obtain the
Fig. 2. Total flattened chromatic dispersion (TFCD) for an Au-PCF with four periods with diameters of air holes d1 , d2 , d3 , d4 and = 1.618 m. The red curve shows the TFCD for five periods with air hole diameters equal to d3 = d4 = d5 = 0.9 m. The black and blue curves show the TFCD for four rings with d3 = d4 = 0.9 m and d1 = d4 = 0.5 m. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
curves of total chromatic dispersion of an Au-PCF using the data of mode index (neff ) or constant of propagation (ˇ), calculated. D() = −
d2 neff cd2
,
(1)
where c is the speed of light in vacuum, neff is the mode index and is the operating wavelength. For the next results, we consider the lattice (pitch), d1 , and d2 , constant and varying at the same time d3 = d4 = 0.85, 0.875, 0.9 and 1.0 m, and only to show the sensitivity of this design, we show some curves where we vary only d4 = d1 = 0.5 m. In Fig. 2 we have three curves with different parameters for an Au-PCF, in the black curve we have four rings, and diameters d3 = d4 = 0.9 m, in this case we have positive dispersion with 0.4455 ± 0.349 ps/(nmkm) in a wide range of wavelengths (1225–1728 nm) and the red curve shows the same parameters including one more air hole ring with diameters equal to d3 and d4 , here we can see that the influence of one more rings does not improve the flattened dispersion or range of wavelengths where it presents a flattened dispersion, but improves the losses for technological applications [13]. The blue curve was obtained by setting the diameter four equal to diameter one (0.5 m), only to show the control we have by varying the diameter of the last ring and we can see how the dispersion curve moves down and presents an ultra-flattened dispersion in communication windows with normal dispersion equal to −0.07425 ± 0.07424 ps/(nm-km) from 1519 nm to 1670 nm and keeping only three different air hole rings in the Au-PCF. Here we show that by only varying the last ring we can have control of the total chromatic dispersion according to the type of dispersion close to ZDW required. In Fig. 3 we can see a high ultra-flattened chromatic dispersion in a wide range and the sensitivity and control we have by varying the two last diameters of rings of the Au-PCF, the solid black and black (square marker) curves show the chromatic dispersion for a four rings structure of an Au-PCF. For the black curve (square marker), we can see that the best result in dispersion can be realized in a wide range of wavelengths, and the variation in dispersion that we have obtained is 1.712 ± 0.022 ps/(nm-km) from 1345 nm to 1530 nm and in the solid black curve we can appreciate a little increase in
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Fig. 3. Total ultra-flattened chromatic dispersion (TUFCD) for an Au-PCF with four periods with diameters of air holes d1 , d2 , d3 , d4 and = 1.618 m. The solid black and black (square marker) curves show the TUFCD for four rings with d3 = d4 = 1.0 m and d1 = d4 = 0.5 m.
dispersion but we have a higher range of wavelengths and closer to ZWD (1.65 ± 0.04 ps/(nm-km) from 1330 nm to 1560 nm). In Fig. 4 we show the control to displace the curve by varying d3 and d4 to normal dispersion or almost center the curve dispersion at the ZDW point. In the blue curve we can see an almost centered curve at ZDW with 0.068 ± 0.488 ps/(nm-km) from 1210 nm to 1780 nm and in the red curve we have a wider range of wavelengths in normal and anomalous dispersion and with a little difference in total dispersion (−0.34 ± 0.66 ps/(nm-km) from 1190 nm to 1830 nm). In the green curve we centered the ultra high-flattened dispersion in 0.00015 ± 0.06153 ps/(nm-km) from 1292 nm to 1555 nm, here we can see that when we move down the curve at the center of the point with ZDW, the total dispersion increases a little but the range of wavelengths where is present this dispersion increases a little too. This flattened-dispersion response
Fig. 5. Electric field profile for the Au-PCF with d1 = 0.5 m, d2 = 0.6815 m, d3 = d4 = 1.0 m and = 1.618 m. (a) Here we used a 1500 nm wavelength to calculate the EFP. (b) Mode propagation inside the core for Au-PCF. Here we show the fundamental mode calculated for 1500 nm and we can see that the higher concentration of field is transmitted inside the core of the Au-PCF.
of the PCF can be utilized in dense-WDM (DWDM) based optical communication systems [19,20]. Here we can conclude that the analysis done for this simple designed PCF achieves best flattened dispersion than reported in literature in this range of wavelength for green curve in different points centered close to ZDW varying the air hole diameter of the second, third or fourth period of Au-PCF and keeping the pitch equal to 1.618 m despite an ultra-flattened dispersion with little variation of chromatic dispersion close to ZDW. Such that if we decrease or increase in 0.018 microns or more the pitch, the ultra-flattened chromatic dispersion increases and moves down or up the ZDW point, for this reason it is important to keep constant the golden ratio parameter of pitch in m ( = 1.618 m). 3. Mode field diameter and electric field profile for Au-PCF
Fig. 4. Total flattened chromatic dispersion (TFCD) for an Au-PCF with four periods with diameters of air holes d1 , d2 , d3 , d4 and = 1.618 m. The green curve shows the TFCD for four periods with air hole diameters equal to d2 = 0.6815 m and d3 = d4 = 1.0 m with ultra high-flattened dispersion centered at the ZDW. The blue and red curves show the TFCD for four rings with d3 = d4 = 0.875 m and d3 = d4 = 0.85 m. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In Fig. 5(a) we show the electric field profile (EFP) calculated for 1500 nm, the propagation mode is distributed in the core of the Au-PCF, the energy of the fundamental mode is concentrated from the center until the first ring of the structure. We can appreciate the smooth distribution of the field with a little evanescence mode inside the core. The EFP has small variations for different diameters of d3 and d4 in the range proposed from 0.85 to 1.0 m and keeping d1 = 0.5 m and d2 = 0.675–0.7 m. From Fig. 5(b) we can see the confinement mode field guided by the core of the Au-PCF for 1500 nm according to the EFP shown in Fig. 5(a). From Fig. 5(b) we can appreciate the mode propagated inside the center of the core for an Au-PCF using the parameters in Fig. 4 for the green curve
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the black (point marker) curve we can see that the MFD distribution with d3 = d4 = 0.875 m varies from 2.8 to 3.8 m with respect to a wavelength in the range of 1200–1800 nm and in the red curve we can see a small variation in the MFD distribution with respect to the blue curve, for this reason we can say that if we keep the first two diameters equal to d1 = 0.5 m and d2 = 0.675–0.7 m and we vary only the two last diameters in the 3rd and 4th periods in the ranges proposed (0.85–1.0 m) to obtain a ultra-flattened dispersion and control the position of the dispersion behavior and the range of wavelength, the MFD is negligible in this range of variation which agrees with Fig. 6. We can appreciate in Fig. 5(b) that the electric field mode is guided inside the core of the Au-PCF because we have a solid core of approximately 2.74 m of diameter according to the pitch proposed for this structure inspired in the relation of nature presented in some leaves, animals and human beings and scaled in m for an Au-PCF structure of pitch. The small values of the MFD and effective mode area for PCFs indicate that these fibers have numerous potential applications in nonlinear optical devices and high performance optical coupling [17,25]. 4. Conclusions
Fig. 6. Mode field diameter and effective area for an Au-PCF. (a) Here we show the MFD for different air hole diameters in the last two periods of an Au-PCF. The MFD is 2.74 m approximately for an Au-PCF, and the variation in the range of wavelength for flattened dispersion is 2.8 m to 3.8 m. (b) Effective mode area agrees with the MFD.
We studied and proposed a simple structure for a PCF with an ultra-flattened dispersion of four rings and golden ratio parameter (in m) of pitch. Here we demonstrated numerically a high sensitivity to control some parameters by only varying the last two periods of the Au-PCF keeping a high-flattened dispersion in a wide range of wavelengths. We showed numerically that it is possible to obtain ultra-flattened dispersion of 0 ± 0.06168 ps/(nm-km) from 1292 nm to 1555 nm. In addition, we analyzed the MFD and showed the electric field distribution for an Au-PCF and we concluded that the Au-PCF for high ultra-flattened dispersion presented a fundamental mode propagation inside the core in a wide range with good confinement which helped us obtain some nonlinear optical processes with a short length Au-PCF. The high sensitivity obtained by varying the two last diameters of an air hole ring makes possible to obtain normal or anomalous dispersion in different ranges of wavelength close to ZWD in the second or third communication windows. Acknowledgements
(d1 = 0.5 m, d2 = 0.6815 m, d3 = d4 = 1.0 m and = 1.618 m). In this case the structure of the fiber concentrates all the energy inside the core in a wide range of wavelengths and we can see that the mode does not present a large evanescence mode outside the center. A mode field diameter is shown in Fig. 6 for two different AuPCFs with four periods keeping the first two diameters of air holes equal to 0.5 m and 0.7 m, 0.6815 m. Here we used Eqs. (2) and (3), where the first one corresponds to a normalized frequency of the PCFs analogy with single index fiber and the second one is the Marcuse formula to evaluate the MFD using the modified Veff number and keeping the cut-off condition for all the range of wavelengths (Veff = 2.405) [21–24]. Veff =
2 a eff
n2co − n2FSM ,
w 1.619 2.879 = 0.65 + + , 6 3/2 aeff Veff Veff
(2) (3)
where aeff is the effective core radius, is the operating wavelength, nco is the core index (for PCFs with solid core we can take the refractive index of silica), nFSM is the effective index of the fundamental space-filling mode and w is the half MFD (called the effective modal spot size), where the background index is assumed to be 1.45. In
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