A novel design of polarization splitter in three-core photonic crystal fibers (PCFs) has been proposed. The three-core PCF consists of two given identical cores with two-fold symmetry separated by a core with high birefringence. The polarization splitter is based on the phenomenon of resonant tunneling. Numerical simulations with a full vectorial beam propagation method demonstrate that it is possible to obtain a 1.9-mm-long splitter with the extinction ratio better than -20 dB and a bandwidth of 37 nm.
©2004 Optical Society of America
Polarization beam splitters, that can split one light into two orthogonal polarization states, are one of essential components in optical fiber communications and integrated photonics. Various types of optical polarization splitters have been proposed [1–5]. As for the optical fiber-based versions, they can be used for a pump combiner in optical amplifiers, a polarization-mode-dispersion compensator, a polarization division multiplexing, and so on, and their operational principle is based on a birefringent-fiber coupler . A disadvantage of conventional fiber-based polarization splitters, however, is that a long coupler length (>a few cm, typically) is required because the birefringence of conventional glass fibers is small.
Recently, photonic crystal fibers (PCFs) , also called microstructured fibers or holey fibers, have attracted significant research attention because they provide extra degrees of freedom in manipulating optical properties. PCFs are usually formed by a central defect region surrounded by multiple air holes with the same diameter in a regular triangular lattice, and the manufacturing technology of PCF such as stack-and-draw technique and the large index contrast between air and silica can easily realize a multi-core PCF [7–9] and a highly birefringent PCF . More recently, Zhang et al. have reported a design principle of realizing compact polarization splitters based on a highly birefringent dual-core PCF . The multi-core PCFs with high polarization dependence have a possibility of the construction of ultra-compact all-fiber polarization splitters. However, the reported polarization splitter based on dual-core PCF, which relies on a large difference in the coupling lengths for the two polarization modes, has relatively low extinction ratio of ~-11 dB.
In this paper, a novel design of polarization splitter in three-core PCFs has been proposed. The three-core PCF consists of two given identical birefringent cores with two-fold symmetry separated by a core with high birefringence. The polarization splitter is based on the phenomenon of resonant tunneling . The PCF is designed to couple only one polarization state in three birefringent cores by adjusting the size of the air holes around the three core regions. Numerical simulations with a full vectorial beam propagation method (BPM)  demonstrate that it is possible to obtain a 1.9-mm-long splitter with the extinction ratio better than -20 dB and a bandwidth of 37 nm.
2. Principle of operation
Figure 1 shows the schematic cross-section of the proposed polarization splitter in three-core PCF. The centers of all air holes are arrayed in a regular triangular lattice with the hole pitch Λ. It consists of two identical birefringent cores A and C separated by another birefringent core B and is formed by using five kinds of air-hole diameter (d 1 to d 5). The cores A and C are formed by combination of the four kinds of air-hole diameter. The large air holes with diameter d 1 are placed in the left and right around the cores A and C. The core B is also formed by combination of the four kinds of air-hole diameter. The large air holes with diameter d 2 are placed in the above and below around the core B.
The operation of this splitter is based on the fact that the difference in the effective refractive indices of the horizontally polarized (x-polarized) and the vertically polarized (y-polarized) modes could be increased by using highly birefringent PCF structures shown in lower panels of Fig. 1. This large difference can be used for polarization selective devices based on the phenomenon of resonant tunneling . In cores A and C, the effective refractive indices of x-polarized mode are smaller than those of y-polarized mode. On the other hand, in core B, the effective refractive index of x-polarized mode is larger than that of y-polarized mode. The structure of core B is so designed that its x-polarized mode is almost resonant with the x-polarized mode of the outer cores A and C. Due to the large difference between the effective refractive index of y-polarized mode in core B and those in cores A and C, the y-polarized mode of core B will be completely nonresonant with those of the outside cores A and C.
The operation of this polarization splitter can also be explained in terms of the supermodes of the three-core directional coupler. If the individual isolated cores of the coupler are single moded, the coupled structure supports three modes, two symmetric and one antisymmetric modes. Let neff ,1, neff ,2, and neff ,3 represent the effective refractive indices of the two symmetric and the one antisymmetric modes, respectively, for each of the polarization states. If we choose the parameters of the PCF to satisfy the condition:
the power transfer efficiency from one outside core to another can be maximized .
In the proposed configuration, the x-polarized modes between the outer cores strongly interact through resonant tunneling, while for the y-polarized mode the interaction is much weaker. Thus, it is possible to choose the parameters as
Choosing a PCF of length L as
the x-polarized mode launched into core A would couple to x-polarized mode in core C, on the other hand, the y-polarized mode launched in core A would mainly exit from core A, where λ is the operating wavelength.
3. Simulation results and discussions
In order to show the applicability of the proposed polarization splitter, we consider a three-core PCF structure in Fig. 1, where the hole pitch Λ=2.0 µm, the relative sizes of air holes are d 1/Λ=0.95, d 3/Λ=0.3, d 4/Λ=0.2, and d 5/Λ=0.5, and the large air-hole diameter d 2 around the core B is taken as a parameter. The high birefringence in cores A and C is achieved by using very large air-hole diameter d 1 and small air-hole diameter d 3. The smaller air-hole diameter d 4 is introduced to attain a short polarization splitter. The air-hole diameter d 2 is so determined that only the horizontally polarized mode in core B is resonant with the horizontally polarized mode of the outer cores A and C. The individual isolated cores of this structure support only x-polarized and y-polarized fundamental modes in 1550-nm wavelength range. As we mentioned in previous section, the necessary condition for obtaining a high-power transfer efficiency between the outer cores is 2neff ,3-neff ,1-neff ,2=0, i.e., the effective refractive indices of the supermodes of the coupler must be equally spaced.
Figure 2 shows the variation of the effective refractive indices of supermodes of the three-core PCF and the value of 2neff ,3-neff ,1-neff ,2 for the x-polarization state as a function of d 2/Λ, where the background silica index is assumed to be 1.45 and operating wavelength λ=1550 nm. In Fig. 2(a), the effective refractive indices of the fundamental modes of the isolated cores A and B, neff ,A and neff ,B, for the x-polarization state are also shown as dashed curves. To obtain the effective refractive indices of the x-polarized and y-polarized supermodes of the three-core structure, a full vectorial modal solver based on finite element method  is used. As seen from the figure, the value of 2neff ,3-neff ,1-neff ,2 for the x-polarized mode becomes zero at d 2/Λ=0.747. At this point the effective refractive indices of x-polarized supermodes are neff ,1=1.415196 and neff ,3=1.414788, and so the coupling length for the x-polarization state is L=1.9 mm. For the y-polarized supermodes of the PCF (neff ,1-neff ,3)y-pol≅1.6×10-5 and (neff ,3-neff ,2)y-pol≅3.947×10-3 showing the high polarization dependence of the coupler.
Next, using a full vectorial BPM based on finite element method  with an anisotropic perfectly matched layers boundary condition , we confirm that the proposed three-core polarization splitter could split two polarization states. The PCF parameters are as follows: Λ=2.0 µm, d 1/Λ=0.95, d 2/Λ=0.747, d 3/Λ=0.3, d 4/Λ=0.2, and d 5/Λ=0.5. The x-polarized and y-polarized fundamental modes at λ=1550 nm are inputted into the core A in Fig. 1 and the beam propagation analysis is preformed.
Figure 3 shows the normalized power variation along the propagation distance in the cores A and C. It is found that the y-polarized mode launched into core A does not couple into the core C, while the x-polarized mode completely couples into the core C and the curve of the power transfer versus the fiber length is very smooth. The separation of two polarization states is achieved at the propagation distance of ~1.93 mm. This result is in good agreement with the coupling length estimated by using (5). The extinction ratio in the three-core PCF is due to uneven spacing of effective indices of three eigenmodes. The extinction ratios, ERA and ERC, are defined, respectively, as
ERA in dB is calculated as 10 times the logarithm of the ratio of the output power of the x polarization in core A and the output power of the y polarization in core A, and ERC in dB is calculated as 10 times the logarithm of the ratio of the output power of the y polarization in core C and the output power of the x polarization in core C. BPM simulation indicates that the extinction ratios are ERA=-30 dB and ERC=-24 dB at λ=1550 nm and the three-core PCF operates as a polarization splitter in 1550-nm wavelength range.
Figure 4 shows the x- and y-polarized mode field distributions versus propagation distance in the polarization splitter. The lights are launched in the core A and the x- and y-polarization fields are separated at the propagation distance of 1.93 mm. The x-polarized mode goes into the core C, while the y-polarized mode remains in the core A.
Figure 5 shows the wavelength dependence of the extinction ratios, ERA and ERC, at a fixed PCF length of 1.93 mm. The extinction ratio ERA becomes minimum at the point where 2neff ,3-neff ,1-neff ,2=0. The bandwidth of -20 dB extinction ratio is almost 37 nm, i.e., from 1536 nm to 1573 nm. The peak value of the extinction ratio ERA and its width with respect to variation of the hole diameter d 2 depends on the hole pitch Λ. Decreasing Λ, the coupling between the modes of individual cores becomes stronger and the resonance would become broader with a lower peak value of the extinction ratio. Thus, the tolerance toward the hole diameter d 2 could be increased by decreasing Λ, although at the cost of decreased extinction ratio.
If the x-polarized and y-polarized modes are separately launched into the cores A and C, respectively, the x-polarized mode couples into core C where the y-polarized mode propagates and the two modes completely combine at the coupling length of the x-polarized mode. In this case the three-core PCF can be used as a polarization combiner.
In this work, we have proposed novel design of polarization splitter in three-core PCF based on the phenomenon of resonant tunneling. The three cores consist of two identical highly birefringent cores with two-fold symmetry separated by another birefringent core. Results from numerical simulations with a full vectorial BPM have been presented. The novel polarization splitter has an extinction ratio better than -20 dB and a bandwidth of 37 nm within a short length of 1.9 mm. The proposed three-core PCF would have a possibility of the construction of ultra-compact all-fiber polarization splitters/combiners.
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