The present paper describes a novel systematic solution to the problem of controlling the chromatic dispersion and dispersion slope in photonic crystal fibers (PCFs), using a structurally-simple PCF with a defected-core. By adjusting the size of the central air-hole defect we can successfully design an ultra-flattened PCF with low confinement losses, as well as small effective mode area. The design strategy is based on the mutual cancellation between the waveguide and the material dispersions of the PCF, by varying the size of the central defected region in the core. The verification of the ultra-flattened chromatic dispersion property of the proposed PCF is ensured with an accurate full-vector finite element method with anisotropic perfectly matched layers. The ultra-flattened dispersion feature, as well as the low confinement losses and the small effective mode area are the main advantages of the proposed PCF structure, making it suitable as a chromatic dispersion controller, dispersion compensator, or as candidate for nonlinear optical applications.
©2005 Optical Society of America
Optical fibers which can transmit the information in the form of short optical pulses over long distances have revolutionized telecommunication industry in the last two decades . A photonic crystal fiber (PCF) on the other hand, is a new class of optical cable that enables light to be controlled in ways not previously possible or even imaginable . As a result of their extra ordinary properties, PCFs have become a pre-eminent method for transmitting information and realizing optical devices .
Index-guiding PCFs, also known as holey fibers have recently attracted considerably attention from the optical community. One of the appealing properties of PCFs is the fact that they can possess dispersion properties significantly different than that of the conventional optical fibers, because their artificially-periodic cladding consisting of micrometer-sized airholes allows the flexible tailoring of the dispersion curves. The task of controlling the chromatic dispersion is a very important problem in designing practical optical communication systems , dispersion controllers , or nonlinear systems .
To achieve nearly-zero ultra-flattened chromatic dispersion in PCFs, several intriguing designs have been proposed so far. Among of them we can distinguish the conventional PCF designs with uniform optimized air-holes [7–9], a PCF with two-defected air-hole rings , and a nonlinear PCF with several kinds of air-hole diameters . Regarding the conventional PCFs [7–9], the small air-hole diameters significantly increase the confinement losses even with large number of air-hole rings. On the other hand the PCF structure proposed in Ref.  supports a higher order mode and so is not a truly single-mode fiber.
The design of simple PCF structure with ultra-flattened chromatic dispersion characteristics, and low confinement losses is an ongoing challenge. Taking all the above circumstances into account, in this paper we propose and numerically characterize a novel type of PCF with both ultra-flattened dispersion characteristics and low confinement losses, without significantly increasing the number of the control parameters. The basic idea relies in introducing a defected air-hole in the core of a PCF structure, which can enhance the waveguide dispersion in conjunction to the material dispersion in purpose to succeed the mutual cancellation between them and thus to obtain nearly-zero dispersion characteristics over a wide wavelength range. The validation of the design is done by using an efficient full-vector finite element method (FEM) with anisotropic perfectly matched layers for accurate modeling of PCFs .
2. Schematic diagram and design guidelines of the defected-core PCF
Consider the schematic cross section of the PCF structure as shown in Fig. 1. It is composed of circular air-holes in the cladding arranged in a triangular array with lattice constant Λ and diameters d. The central core region is perturbed by including an extra air-hole with diameter dc. The host material is regular silica. The total number of air-hole rings was chosen to be four in order to simplify as much as possible the structural composition of the PCF. To accurately simulate this PCF architecture we have adopted an efficient full-vector FEM  with anisotropic perfectly matched layers for predicting with high accuracy all the propagation characteristics of the waveguide. The insertion of the extra air-hole in the core region is actually the intriguing concept of the present design, and according to our extensive literature research is a novel idea and will be reported for the first time. The existence of the central defected air-hole has the function to control the waveguide dispersion properties of the fiber as will be demonstrated later on. Our design technique starts with the observation that the total dispersion in a PCF structure can be well approximated using the following relation :
where D(λ) is the total dispersion of the PCF, Dw(λ) is the waveguide dispersion, and Dm(λ) is the material dispersion which can be obtained using the Sellmeier’s equation. In order to obtain nearly-zero total dispersion, we can see from Eq. (1) that if we are able to design the PCF in such a way that can exhibit a waveguide dispersion nearly opposite to that of the material dispersion, namely Dw(λ) ≅ -Dm(λ) over a finite number of frequencies , we can partially fulfill the nearly-zero dispersion requirement. In Fig. 2 we plot the calculated normalized waveguide dispersion response Dw(λ)Λ for various incremental values of the design parameter, dc/Λ = 0 (red line), dc/Λ = 0.1 (green line), dc/Λ = 0.2 (blue line), dc/Λ = 0.3 (cyan line), dc/Λ = 0.35 (purple line), and dc/Λ = 0.4 (brown line), where the silica index is assumed to be 1.45 for calculating waveguide dispersion. Specifically Fig. 2(a) shows the waveguide dispersion curves at d/Λ=0.65, Fig. 2(b) shows the dispersion curves at d/Λ=0.70, Fig. 2(c) shows the dispersion curves at d/Λ=0.75, and Fig. 2(d) at d/Λ=0.80. From these results we can clearly see that the continuous increment of the defected air-hole size in the core, down-shifts the waveguide dispersion to negative values. On the other hand the increment in the size of the air-hole diameters d in the cladding, up-shifts the waveguide dispersion curves. In Fig. 2(c) we additionally plot the material dispersion curves which are monotonically increasing concave smooth functions of the wavelength (in most of the infrared region), corresponding to different lattice constants, of Λ=1.5 μm (dashed red line), Λ=2.0 μm (dashed green line), and Λ=2.5 μm (dashed blue line). The enlarged inset picture in Fig. 2(c) shows the obtained anti-symmetric curves of the waveguide and material dispersions and as we can observe under the relation (1), the algebraic sum of these lines can lead to nearly-zero total dispersion as required. Then by a judicious choice of the three design parameters, that is the lattice constant Λ, the air-hole diameters in the cladding d, and the diameter dc of the defected air-hole in the core, we can further make some micro-adjustments in the waveguide dispersion and it’s dispersion slope, in purpose to succeed having ultra-flattened total dispersion curves in some particular frequency regime (for example in the telecommunication window). The physical mechanism of the proposed design procedure can be explained as follows: the continuous enlargement of the defected air-hole in the central silica-region reduces the portion of the material in the core and as a result there is a compensation of the inherent dispersion of the silica. In Fig. 3 we show the detailed impact of the design parameters Λ, d, dc, to the total dispersion curve of the PCF. Notice that the results in Fig. 3 were obtained not by the relation (1) but using the exact definition of the total chromatic dispersion as:
where in Eq. (2) neff is the effective index of the fundamental mode and the material dispersion based on Sellmeier’s equation has been taken into account explicitly in the effective index of the PCF. Specifically Fig. 3(a) shows the impact of the micro-tuning of the lattice constant Λ to the total dispersion obtained using Eq. (2), for Λ=2.0 μm (blue line), Λ=2.05 μm (red line), and Λ=2.1 μm (green line). Figure 3(b) shows the impact of the diameter variation in the cladding d of the PCF, for d/Λ=0.72 (blue line), d/Λ=0.73 (red line), d/Λ=0.74 (green line), while Fig. 3(c) shows the impact to variation of the defected air-hole diameter in the core of the PCF, for dc/Λ=0.276 (blue line), dc/Λ=0.279 (red line), and dc/Λ=0.282 (green line). According to these numerical results, there exist an optimized set of design parameters that is Λ, d, dc which can lead to ultra-flattened total chromatic dispersion. These optimized values have been obtained through our numerical simulations to be Λ=2.05 μm, d/Λ=0.73, and dc/Λ=0.279. Thus the optimized chromatic dispersion responses of the PCF under consideration correspond to the red lines in Fig. 3. To quantify the performance of our proposed PCF we have computed the ultra-flattened chromatic dispersion to be 0.2±0.2 ps/km/nm, from 1.14 μm to 1.7 μm. From Fig. 3(c) a typical variation of the design parameter dc by a ±1 % will result in a change of the chromatic dispersion curve of about ±2 ps/km/nm. This result shows that the newly proposed structure is indeed sensitive to the design parameter dc. The same observation however is true for all nonlinear PCFs (due to their small effective mode area), and thus the feasibility of our PCF structure is almost identical to the feasibility of previously proposed structures [5, 11]. The advantage of our structure is the fact that is much simpler from the design point of view, in comparison to previous designs because it contains less number of design parameters.
3. Confinement losses and effective mode area of the proposed PCF structure
We have shown already that the proposed PCF structure can exhibit ultra-flattened dispersion characteristics. In practical applications however only the near-zero dispersion characteristics may not be enough for justifying the usefulness of the fiber. Low confinement losses and/or small effective mode areas are needed for some particular applications like nonlinear ones. We will show that our proposed PCF structure has in addition both the above mentioned characteristics, which are low confinement losses as well as the small effective mode area. In Fig. 4 we plot the calculated effective mode area (blue line) and the leakage loss (red line) as a function of the wavelength λ, for the optimized design parameters Λ = 2.05 μm, d/Λ = 0.73, and dc/Λ = 0.279. From the results in Fig. 4 we can clearly observe the remarkably low confinement loss, that the proposed PCF with only four air-hole-rings exhibits (0.013 dB/km at wavelength of λ = 1.55 μm), and in addition we can see the relatively low effective mode area (6.1 μm2 at wavelength of λ = 1.55 μm). The leakage loss of the higher order mode was calculated to be larger than 10 dB/km for wavelengths above λ =1.5 μm. So in the telecommunication window this PCF effectively operates as a single mode fiber. Finally from the electric field distribution of the x-polarized mode in Fig. 5 at a wavelength of λ = 1.55 μm, we can see the strong confinement of light in the core of the PCF. However the existence of the defected air-hole in the core slightly reduces the effective core index and as a result the field lines penetrate the cladding more strongly in comparison with the non-defected core PCFs. At this point we have to resolve an issue regarding possible low coupling efficiency of the proposed structure. The fact that our proposed PCF exhibits small effective mode area with an air hole in the middle of the core could result in low coupling efficiency when splicing to standard single mode fibers (SMFs). This issue however may be effectively solved by splicing the newly proposed device with a high numerical aperture fiber . This method has been experimentally verified to result in high coupling efficiencies between a SMF and a nonlinear PCF (with small effective mode area).
To summarize this paper we have demonstrated an outstanding technique for controlling the chromatic dispersion in PCFs by introducing a defected air-hole in the core of the structure. Our proposed PCF architecture is significantly simpler than other structures proposed so far for controlling the chromatic dispersion and at the same time ultra-flattened dispersion characteristics have been obtained over a wide frequency range. Additionally our proposed PCF architecture shows low confinement losses as well as small effective mode area, which are novel properties in an ultra-flattened dispersion design. The main conclusion of this systematic approach is that with a modest number of design parameters we could fully-tune and optimize the chromatic dispersion properties of the PCF. Our architecture is suitable for applications as a chromatic dispersion controller, dispersion compensator, or as candidate for nonlinear optical systems because of its small effective mode area. Other PCF architectures which can exhibit ultra-flattened dispersion characteristics with low confinement losses based on the inclusion of elliptical air-holes in the cladding are currently under consideration.
References and links
1 . J. A. Buck , Fundamentals of Optical Fibers , Wiley-Interscience ( 2004 ).
3 . J. C. Knight , T. A. Birks , P.St.J. Russel , and D. M. Atkin , “ All-silica single-mode optical fiber with photonic crystal cladding ,” Opt. Lett. 21 , 484 – 485 ( 1996 ). [CrossRef]
4 . M. D. Nielsen , C. Jacobsen , N. A. Mortensen , J. R. Folkenberg , and H. R. Simonsen , “ Low-loss photonic crystal fibers for transmission system and their dispersion properties ,” Opt. Express 12 , 1372 – 1376 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-24-1372 . [CrossRef] [PubMed]
5 . K. Saitoh , M. Koshiba , T. Hasegawa , and E. Sasaoka , “ Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion ,” Opt. Express 11 , 843 – 852 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-08-843 . [CrossRef] [PubMed]
6 . V. Finazzi , T. M. Monro , and D. J. Richardson , “ Small core silica holey fibers: Nonlinearity and confinement loss trade-offs ,” J. Opt. Soc. Am. B 20 , 1427 – 1436 ( 2003 ). [CrossRef]
7 . A. Ferrando , E. Silvestre , J. J. Miret , and P. Andres , “ Nearly zero ultraflattened dispersion in photonic crystal fibers ,” Opt. Lett. 25 , 790 – 792 ( 2000 ). [CrossRef]
8 . A. Ferrando , E. Silvestre , P. Andres , J. J. Miret , and M. V. Andres , “ Designing the properties of dispersion-flattened photonic crystal fibers ,” Opt. Express 9 , 687 – 697 ( 2001 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687 . [CrossRef] [PubMed]
9 . W. H. Reeves , J. C. Knight , P. St. J. Russell , and P. J. Roberts , “ Demonstration of ultra-flattened dispersion in photonic crystal fibers ,” Opt. Express 10 , 609 – 613 ( 2002 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687 . [PubMed]
10 . T. L. Wu and C. H. Chao , “ A novel ultraflattened dispersion photonic crystal fiber ,” IEEE. Photon. Technol. Lett. 17 , 67 – 69 ( 2005 ). [CrossRef]
11 . K. Saitoh and M. Koshiba , “ Highly nonlinear dispersion-flattened photonic crystal fibers for supercontinuum generation in a telecommunication window ,” Opt. Express 12 , 2027 – 2032 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2027 . [CrossRef] [PubMed]
12 . K. Saitoh and M. Koshiba , “ Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers ,” IEEE J. Quantum Elencron. 38 , 927 – 933 ( 2002 ). [CrossRef]
13 . D. Davidson , Optical-Fiber Transmission ( E. E. Bert Basch , ed., Howard W. Sams & Co , 1987 ).
14 . H. C. Nguyen , B. Kuhlmey , M. J. Steel , C. Smith , E. Magi , R. C. McPhedran , and B. Eggleton , “ Leakage of the fundamental mode in photonic crystal fiber tapers ,” Opt. Lett. 30 , 1123 – 1125 ( 2005 ). [CrossRef] [PubMed]