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A particular photonic crystal fiber (PCF) designed with all circle air holes is proposed. Its characteristics are studied by full-vector finite element method (FEM) with anisotropic perfectly matched layer (PML). The simulation results indicated that the proposed PCF can realize high birefringence (up to 10−2), high nonlinearity (50W−1·km−1 and 68W−1·km−1 in X and Y polarizations respectively) and low confinement loss (less than 10−3dB/km at 1.55um wavelength).
© 2015 Optical Society of America
1. Introduction
In recent years there has been an increasing interest in photonic crystal fibers (PCFs) [1–4]. Owning to their flexibility design for the cross section, PCFs can realize particular properties such as high birefringence, high nonlinearity, ultra-flatten dispersion, large effective mode area, endlessly single mode, and etc [5–10]. PCFs specially designed for high birefringence can be used as polarization maintaining fibers or single-polarization-single-mode fibers in long distance communications, optical fiber sensing and special laser systems such as fiber gyroscopes and polarization-sensitive optical modulators [11,12]. Generally speaking, there are two kinds of birefringence, namely, geometrical birefringence [13] and stress birefringence [14]. So far, several PCFs have been reported with high birefringence [15–18]. Among them, all elliptical air holes PCFs have been designed to enhance the birefringence up to the order of 10−2 [17,18], but these PCFs suffer from poor light confinement and high propagation loss. To overcome the weakness of poor mode confinement, PCFs with hybrid air holes, i.e. contain circle air holes in the cladding and elliptical air holes in the core area, have been proposed [19–21]. High birefringence and low confinement loss can be achieved for this kind of PCFs, however, it is rather challenging to implement these hybrid designs in the present fabrication processes.
In this study we show high birefringence, high nonlinearity and low confinement loss can be achieved simultaneously by a design of all circle-air holes PCF. The complex hybrid air holes design is avoided without degradation of performance, and, since all air holes are circular, this PCF could be fabricated by the stack and draw method or the performs drilling method. The design is illustrated in Fig. 1. Optimum parameters are selected to make sure that the fiber is single-mode above 1.45μm, with modal birefringence of as high as 2.2 × 10−2, nonlinearity of 50W−1·km−1 and 68 W−1·km−1 in the two perpendicular polarizations, and confinement losses of typically less than 10−3dB/km at 1.55μm for both polarizations.
2. Design and optimization of the PCF structure
The cross section of the proposed PCF design is shown in Fig. 1, where three small air holes in the core region and two larger air holes in the first ring of the cladding together form an anisotropic core area. The rest of the cladding is composed of large air holes arranged in a triangular array in silica background. The center-to-center spacing between the air holes is Λ. The diameter of air holes in the cladding is d. The two bigger air holes with diameter d1 are used to enhance modal birefringence of the PCF. The air holes in this design are relatively large compared to most traditional single-mode PCF. The special feature of this PCF structure is the appearance of three small air holes in the core area. Attributed to this special feature, a decreased confinement loss and an enlarged single mode range of operation is realized for the fiber. Among the three unusual air holes the upper one and the lower one are identical in size with diameter d2, while the middle one is slightly larger with diameter d3 and shifted horizontally by a distance L along the X-axis. The two small air holes are placed on the Y-axis with a distance H from the origin. This arrangement establishes an eccentric shape of the core and a reduced area of silica, which result in both high birefringence and high nonlinearity. This design maintains the high level performance of the hybrid air holes design [19–21] while slightly reduces manufacturing difficulty.
Nonlinearity and birefringence of the PCF depend on concentration and asymmetry of the fiber mode, which is controllable by size and position of the air holes in the core. The commercial software COMSOL Multiphysics, which is based on the full-vector finite element method (FEM) with the perfectly matched layer (PML) boundary condition, is adopted for the analyses of the structure. The field distribution and the modal effective indices are calculated for different modes. After the complex modal effective index (neff) is obtained, the modal birefringence B is determined by the absolute difference of neff for the two polarizations, while the confinement loss is represented by the imaginary part of the neff [11]. The nonlinear coefficient γ is given by Eq. (1), where λ is the operational wavelength, n2 = 3.2 × 10−20m2/W [24] is the nonlinearity index of silica and mode effective area Aeff is given by Eq. (2). In the calculations the refractive index of silica is determined by the Sellmeier equation [25].
2.1. Effect of air hole diameter in the cladding on birefringence and nonlinearity
To focus on the impact of air hole diameter on the birefringence and the nonlinearity, we change the value of d, the diameter of the holes in the cladding, while fix the other parameters to λ = 1.55μm, Λ = 2μm, d1 = d, d2 = 0.36Λ, d3 = 0.465Λ, L = Λ/8, H = 0.43Λ. Figure 2 shows dependence of the mode effective refractive indices, the modal birefringence, and the nonlinear coefficient as d increases. As illustrated by Fig. 2(a), the effective refractive index of the Y-polarization is always larger than that of the X-polarization. This phenomenon can be understood as that there are more air in the X direction owing to the shape of the core. As d increases, both effective refractive indices of the X and Y polarized modes decrease, while both nonlinear coefficients of the X and Y polarized modes increase. This is because larger d corresponds to larger filling fraction of air holes, in turn means lower average refractive index of the cladding. As a consequence, effective refractive indices of both the X and Y polarized modes decrease. The increase of hole diameter in the cladding nibbles area of the core, and small core area means high concentration of light. Consequently, nonlinear coefficients of both the X and Y polarized modes increase. Birefringence of the fiber increases as d increases. This may also be explained by the concentration of mode fields in the core area as d increases. Since the core is anisotropic, the more field shrinks back to the core, the higher birefringence the mode becomes. As d = 0.86Λ value of the birefringence reaches 0.016.
Then we fix d = 0.86Λ and further increase d1. The other parameters are kept the same as before. Dependence of the modal birefringence and the nonlinear coefficient as d increases is presented in Fig. 3(a). It's found that the birefringence of the PCF further increases from 0.0163 to 0.022, indicating the two special air holes in the first ring of the cladding have strong influence on asymmetry of the fiber modes. As shown in Fig. 3(a), the nonlinear coefficient of the Y-polarized mode increases a bit. On the other hand, curve of the nonlinear coefficient of the X-polarized mode is almost flat. This can be explained by referring to the electric field distributions of the X and Y polarized modes plotted in Figs. 3(b) and 3(c). The left larger air hole in the first ring squeezes the core in the X direction. For the X-polarized mode boundary conditions on the air hole edges allow the light field to penetrate into the air holes, so concentration of light remains almost constant. On the other hand, boundary conditions force field of the Y-polarized mode to concentrate within the shrunk core as d1 increases so that nonlinear coefficient of the Y-polarized mode increases with d1.
Fig. 3 (a) Modal birefringence and nonlinear coefficient of the proposed PCF versus d1, and electric field distributions of respectively the X (b) and the Y (c) polarized modes at d1 = 0.95Λ.
2.2. Effect of air hole diameter in the core on birefringence and nonlinearity
Having decided diameter of the air holes in the cladding, we now consider the air holes in the core. The birefringence and nonlinearity can be adjusted by changing the holes. Figure 4 shows the modal birefringence and the nonlinear coefficient of the Y polarization as functions of d3 for different d2. The other parameters are set to λ = 1.55μm, Λ = 2μm, d = 0.86Λ, d1 = 0.95Λ, L = Λ/8, H = 0.43Λ. The complex modal effective index neff of both the X and Y polarizations decrease with d3. Because the X-polarized mode is more sensitive to change in d3, nx eff decreases at faster rate than ny eff, so the resulting birefringence increases almost linearly with d3. For fixed d3 the birefringence and nonlinearity increase with d2. Confinement of the PCF also increases with d2 and d3. However, the value of d3 can't increase without a limitation. It is bounded by the restriction set by technology of fabrication on thickness of silica wall between two air holes. A compromise between performance and fabrication has to be made. After some considerations we choose d2 = 0.36Λ and d3 = 0.465Λ as the optimized values.
2.3. Effect of distribution of the air holes in the core on birefringence and nonlinearity
The effect of air hole positions in the core on birefringence and nonlinearity is studied for given diameters of the air holes in the cladding and core. According to the above discussions, restriction on minimum thickness of silica wall between two air holes must be followed. To avoid overlap of air holes in the core with those in the cladding, H and L cannot be set to large values simultaneously. In the following study we fix distance of the two small air holes in the core to H = 0.43Λ and vary offset L. The other parameters are set to λ = 1.55μm, Λ = 2μm, d = 0.86Λ, d1 = 0.95Λ, d2 = 0.36Λ, d3 = 0.465Λ, H = 0.43Λ. Figure 5 shows the birefringence increases monotonically as L decreases. For the same reason as that discussed in the previous section for Fig. 3(a), while the nonlinear coefficient for the X polarization decreases slowly with the change of L, curve of the nonlinear coefficient of the Y polarization is rather flat. The largest nonlinear coefficient are respectively 53.8 W−1·km−1 and 69.2 W−1·km−1 for the two polarizations. Due to the enhanced asymmetry shape of the core, modal birefringence can reach as high as 0.024.
It is well known that in a traditional PCF higher order modes will set in as air filling fraction of the cladding becomes large. In our case, this problem is relieved by the presence of the three air holes in the core area. Figure 6 shows the dispersion curves of the fundamental core modes and the fundamental-space-filling (FSM) mode for the optimum PCF structure, together with the dispersion curves of the Y-polarized second-order core mode for different value of L. The other parameters are Λ = 2μm, d = 0.86Λ, d1 = 0.95Λ, d2 = 0.36Λ, d3 = 0.465Λ, H = 0.43Λ. When the dispersion curve of a mode falls below the curve of the FSM, it is cutoff. In the new proposed fiber the Y-polarized second-order mode has the highest effective index among all the higher order modes. If it is cutoff all higher order modes are cutoff. So it is enough to investigate the Y-polarized second-order mode for the determination of single-mode and multimode ranges. It is revealed by Figs. 5 and 6 that smaller L leads to higher birefringence, but it also delays cutoff of higher order modes. There exists an optimum value of L for a given application. In order to ensure high birefringence and high nonlinearity, and also meet the single mode range requirement, we choose the value of L to be Λ/8. This leads to a cutoff wavelength of 1.45μm for the Y-polarized second-order mode, which is very good for our rather large air filling fraction of the design.
Fig. 6 Dispersion curves of the modes of the proposed fiber. For the 1st-order and the FSM modes, and for the 2nd-order modes L is marked on the figure as a parameter. The other parameters used for the calculations are Λ = 2μm, d = 0.86Λ, d1 = 0.95Λ, d2 = 0.36Λ, d3 = 0.465Λ, H = 0.43Λ.
2.4. Other features of the proposed PCF
Confinement loss of a fiber is another important property for light guiding. It limits the total transmission distance in real fiber communication systems. Confinement loss Lc of PCFs can be calculated by [21]
where Im[neff] is the imaginary part of the effective refractive index. The propagation loss of high order modes is much larger than that of fundamental mode. Furthermore, with the optimum structure parameters the single mode range starts from 1.45μm in our design. Therefore, we only need to focus on the fundamental mode in the following.Confinement losses of both polarizations in the proposed PCF is plotted in Fig. 7 as functions of the number of air hole rings N surrounding the core. It decreases rapidly with N. Since relatively large air holes are used here, light is confined quite well in the core. The presence of the three air holes in the core area also helps the situation, so the confinement loss of the new fiber is much less than that of traditional fibers. The confinement loss of the X-polarized mode is lightly larger than that of the Y-polarized mode. This can be explained by the observation of Figs. 3(b) and 3(c) for the mode fields, where it is shown that the X-polarized mode penetrates the air holes further so is less confined within the core. The confinement loss of the X-polarized mode is about 0.33dB/km for a three-ring PCF and 0.0092dB/km for a four-ring PCF. The confinement losses in both polarizations reduce to below 10−4 as N is 5 or bigger. From simulations it is observed that as the number of rings N surrounding the core changes from 3 to 6, the effective refractive indices, the birefringence, and the nonlinear coefficient of the fundamental modes are almost unchanged. From considerations of fabrication and cost, N = 4 is enough for most applications. From these discussions an optimized structure is arrived with parameters Λ = 2μm, d = 0.86Λ, d1 = 0.95Λ, d2 = 0.36Λ, d3 = 0.465Λ, L = Λ/8, H = 0.43Λ, and N = 4. The new proposed PCF can be fabricated by the stack and draw method or the performs drilling method [22,23]. In recent years there have been some even larger air filling fraction with complex design fabricated [23,26,27], the new proposed design wouldn't be proven a problem.
When the fiber is not drawn to the precise parameters during fabrication process, the performance of PCFs degrades. The sensitivity of the performance to fluctuations of structure parameters is analyzed to reveal robustness of the design. Figures 8(a) and 8(b) show the birefringence and nonlinear coefficient as the parameters vary up to ± 5% around the designed values. From the curves it is seen that ± 5% variation in center-to-center spacing Λ causes ± 5.2% change in the birefringence and ± 3% change in the nonlinearity. Further investigations show that ± 5% shift in the center holes' position in either X direction or Y direction leads to at most ± 6% change in the birefringence and ± 4% change in the nonlinearity. Our design is demonstrated to have a good tolerance of fabrication errors.
Fig. 8 (a) Modal Birefringence and (b) nonlinear coefficient in Y polarization as functions of wavelength for different center-to-center spacing Λ.
Group velocity dispersion of this PCF is evaluated based on the mode effective refractive index, as presented in Fig. 9 for X and Y polarizations as functions of wavelength. In the communication wavelength band both polarizations experience anomalous dispersion. Due to the rather big negative value and almost constant slope, the proposed PCF could be used for dispersion compensation. It is worth mentioning here that if material with higher refractive index, e.g., AsSe2 glass, is used for the background instead of silica, the birefringence and the nonlinear coefficients of the proposed PCF can be further increased [28].
3. Conclusion
To summarize, a high birefringent photonic crystal fiber is proposed and analyzed by the full-vector finite element method. The modal birefringence is 0.022 at 1.55μm with very low confinement losses. The nonlinear coefficients are respectively 50W−1·km−1 and 68W−1·km−1 for X and Y polarizations. Furthermore, single mode range of the fiber is enlarged by the presence of small air holes in the core area. Other properties such as group velocity dispersion are discussed, and tolerance of fabrication is also assessed. The new proposed PCF may found applications in optical fiber sensing [29], polarization maintaining transmission systems [30], and super continuum generation for frequency metrology [12, 31].
Acknowledgment
This work was supported by the National Natural Science Foundation of China (No. 60588502, 60607005, and 60877033).
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