We have proposed directional couplers operated by resonant coupling in all-solid photonic bandgap fibers structure which consist of a cladding with an array of high-index rods in silica background, two cores formed by omitting two rods, and some defect rods introduced by reducing the diameter of the high-index rods between the cores. The resonant effect induced by the avoided crossing between core-guided supermodes and defect-guided modes significant decreases the coupling length. The directional couplers proposed in this paper are almost polarization independent and have potential application in realizing integrating all-fiber communication devices.
© 2007 Optical Society of America
Photonic bandgap fibers (PBGFs) which confine light into core by the photonic bandgap (PBG) effect of the 2D photonic crystal cladding have attracted great interest due to their special guidance mechanism and potential applications in nonlinear optics, high power guidance, optical communication devices, and so on [1, 2]. Recently, some theoretical researches on multi-core photonic bandgap fibers (PBGFs) have been presented, which show these fibers can be designed to have some remarkable coupling properties, such as decoupling effect  and extremum points in coupling length . Skorobogatiy et al. show the directional coupler can be realized by resonant effect in hollow core PBGFs which has two non-proximity cores . Based on this design, the idea of designing complex all-fiber communication device in a single draw process is proposed [5, 6].
All-solid PBGFs commonly are composed of the a two-dimensional array of high-index germanium-doped silica rods in a silica background, have become more attractive because they can not only exhibit the unique properties of PBGFs, but also are easier to fabricate and couple with conventional fibers than the hollow core PBGFs [7,8]. Moreover, solid core and germanium-doped cladding make a possibility to integrate the rare-earth doped amplifier and FBG in this fiber. Recently, dual-core all-solid PBGFs has been theoretically and experimentally studied and some special coupling properties are found in it, e.g. maxima and minima in coupling length, complete decoupling of the cores, and an inversion of the usual ordering of supermodes . Therefore, all-solid PBGF would be a good choice to realize the all-fiber communication devices which integrate more interesting functionality. In this paper, we numerically present the design of directional coupler operated by resonant coupling based on all-solid PBGFs. The directional couplers consist of a cladding with an array of high-index rods in silica background, two cores formed by omitting two rods, and some defect rods introduced by reducing the diameter of the high-index rods between two cores. Resonant effect obviously decreases the coupling length exactly when the distance between two cores is long in our design. The directional couplers proposed here are almost polarization independent.
2. Structures and computation method
Figure 1 shows the resonant coupler of all-solid PBGFs proposed in this paper. The background material is silica with index is 1.457 and the circles denote the raised-index rods which formed by germanium-doped rods with index 1.47288. The material dispersion is neglected since it has no significant impact on our results. The cores are formed by missing two rods in the perfected photonic crystal cladding structures. The rod diameter d is 0.7Λ, where Λ is rod pitch. Other defects (brown circles) are introduced by decrease the diameter of the raised-index rods between the cores. The diameter of the defect rods dd=0.75d. For the structure in Figs. 1(a), 1(b), and 1(c) the distance of two cores Dc is 8Λ, 4Λ, and 6Λ, respectively. For more general case, two cores can be separated by more defect rods. It is known that all-solid PBGF have many bandgap which correspond to some discrete transmission windows. In this paper, we just consider the second bandgap since the transmission window in this bandgap covers the communication wavelength when the diameter of the cores of the PBGFs matches with that of conventional fiber. This bandgap divides with other bandgaps by the LP11-like and LP21-like rod modes. The coupling length Lc is the distance along the fiber in which there is total transfer of power from one core to the other:
where ni even and ni odd are the effective index of i-polarized even and odd supermodes, respectively, k=2π/λ is the vacuum wave vector and λ is the wavelength in vacuum.
We obtained the dispersion curves of PBG edges in the 2D perfect photonic crystal cladding by full-vector plane-wave expansion method using the MIT package . Then the effective index n eff and field distribution of the eigenmodes in all-solid PBGFs were calculated by full-vector finite element method (FEM). The geometric symmetry of PBGFs allows just one quarter of the structure to be considered in the FEM simulation.
3. Results and discussion
To more easily understand the resonant couple in the structure in Fig. 1(b), we firstly consider the cases in two kinds of single core PBGFs which consist of the same cladding structure as that in Fig. 1(b). For the first structure of the single core PBGFs shown in the top-left inset of Fig. 2, the core is formed by missing a rod; for the second one shown in the bottom-right inset of Fig. 2, the core is a smaller rod with the diameter is 0.75d. In the second bandgap, two degenerate LP01-like modes for x- and y-polarization are found in the first structure, while the LP11-like modes are found in the second one. As shown in Fig. 2, the dispersion curves of the LP01- and LP11-like modes intersect each other in the center of the bandgap. At the point of intersection, these modes in two structures have the same effective index.
Next, we study the structure in the Fig. 1(b). In this case, due to the presence of two cores, the LP01-like mode for each polarization in cores splits into two supermodes, which, according to the parity of the mode field with respect to the geometric symmetry axis between two cores, are referred to as the even and odd core-guided supermodes. As calculated previously, the defect rod located on the symmetric axis between two cores can support LP11-like modes. Although the linearly polarized (LP) representation of the eigenmodes is valid for the single core PBGFs, more accurate vector representation should be used when we consider the interaction between core-guided supermodes and defect-guided modes. For the vector representation, the LP11-like modes consists of four nearly degenerate modes: TM01-like, TE01-like, and two HE21-like modes. Figure 3 shows the effective index of the x-polarization dominating modes in the structure of Fig. 1(a). These modes include two core-guided supermodes (even supermode and odd supermode), and two defect-guided modes (TM01-like mode and HE21-like mode). Because the odd supermode and HE21-like mode have same symmetry, the avoided crossing occurs between the dispersion curves of these modes. Figure 4 shows the mode fields at the points denoted in the Fig. 3. Figures 4(a) and 4(b) are the mode fields of even supermode and TM01-like mode, respectively. Since no avoided crossing effect occurs, the fields of even supermode and TM01-like mode mainly locate on the core region and the defect rod region in the whole transmission windows, respectively. The dispersion curves of the odd supermode and HE21-like mode have two branches in Fig. 3. As shown in the Fig. 4, the mode transfers from odd supermode to HE21-like mode on the upper branch and from TM01-like mode to odd supermode on the lower one. The effective index of odd supermode is higher in low frequency region and lower in high frequency region than that of even supermode. Due to the avoided crossing effect, the spacing between the even and odd supermode increase near the resonant point where the spacing between the even supermode and the two branches for the mixing of odd supermode and HE21-like mode is same. As a result, the coupling length obviously decreases at the resonant point. For y-polarization dominating case, the properties of modes are almost same as that of x-polarization dominating case except the TM01-like mode is replaced by the TE01-like modes.
The coupling length of the structure shown in Fig. 1(b) is shown in Fig. 5. For the sake of comparison, the coupling length without the resonant defect was also calculated. From Fig. 5, one can see that the curves of coupling length are significantly different for the fiber with and without defect rod, although the difference between two structures is slightly. For the case without the defect rod between two cores, the maximum are obtained in the middle of the bandgap where the core modes are more tightly confined than that at the edges of the bandgap . Whereas when the defect rod is introduced, power can transfer between the cores through the rod and the coupling length would get minimum at the resonant point. As a result, the coupling length becomes observably shorter than that in the fiber without resonant defect. Note that due to low index contrast, the difference of the coupling length between x and y polarization is very small.
Figure 6 shows the normalized waveguide group velocity dispersion (GVD) for even and odd supermodes of the PBGF shown in Fig. 1(b). To compare it, the waveguide GVD of fundamental mode of the PBGF with a single core by missing a rod is also shown in Fig. 6. The curve of waveguide GVD for even supermode is similar to one for single core PBGF. Whereas due to the avoided crossing effect, large normal GVD occurs for the lower branch of odd supermode and large abnormal GVD for upper branch of odd supermode.
When the distance of two cores increases, more defect rods are introduced between two cores, as shown in Figs. 1(a) and 1(c). In this case, some supermodes are formed by the HE21-like modes in each defect rod. These defect-guided supermodes are also even or odd regarding the symmetry axis between two cores. The avoided crossing effect will take place between these defect-guided supermodes and the core-guided supermodes when they have same parity. Figure 7 shows the coupling length in the PBGFs with different Dc =4Λ, 6Λ, and 8Λ as a function of the normalized frequency. Since the difference of coupling length for two polarizations is very small, we just show the case of x-polarization dominating modes. For each polarization, the number of the defect supermodes formed by HE21-like modes equals to the number of the defects. As a result, the number of the resonant point increases as the number of the defect rods increases. In the case of Dc =6Λ, three defect rods are introduced and two resonant points are observed, while in the case of Dc =8Λ, five defect rods are introduced and four resonant points are observed. The reason for the number of resonant points is less than the number of defect rods is that some defect-guided supermodes not intersect with the core-guided supermodes in the bandgap range. The coupling length rapidly increases near the edge of high frequency of transmission windows for both Dc =6Λ, and 8Λ because no resonant coupling occurs in this region. Whereas, in other region of the transmission windows, the coupling lengths fluctuate in the almost same range for Dc =4Λ, 6Λ, and 8Λ due to the resonant effect.
In conclusion, we proposed directional couplers operated by resonant based on all-solid PBGFs in which two cores are linked by some defect rods. The resonant takes place due to avoided crossing effect between core-guided supermodes and defect-guided modes. The resonant effect significantly decreases the coupling length, which make a possible to realize more complex communication devices. The structure proposed in this paper is easier fabrication than its counterpart based on hollow core PBGFs. Some important fiber component in communication system, such as FBG and fiber amplifier, could be introduced in the structures for integrated all-fiber devices. These subjects are now under consideration.
This work was supported in part by the National Basic Research Program of China (973 project, Grant No. 2003CB314906), the 863 National High Technology Program of China (Grant No. 2006AA01Z217), the National Natural Science Foundation Projects of China (Grant No. 10674074 and 10674075), Tianjin Natural Science Foundation Project of China (Grant No. 06YFJZJC00300), China Postdoctoral Science Foundation (Grant No. 20060400687).
References and links
1. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]
3. Z. Wang, G. Y. Kai, Y. G. Liu, J. F. Liu, C. S. Zhang, T. T. Sun, C. Wang, W. G. Zhang, S. Z. Yuan, and X. Y. Dong, “Coupling and decoupling of dual-core photonic bandgap fibers,” Opt. Lett. 30, 2542–2544 (2005). [CrossRef] [PubMed]
4. J. Laegsaard, “Directional coupling in twin-core photonic bandgap fibers,” Opt. Lett. 30, 3281–3283 (2005). [CrossRef]
7. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. S. Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13, 2503–2511 (2005). [CrossRef] [PubMed]
8. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. S. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express 13, 309–314 (2005). [CrossRef] [PubMed]