We study the birefringence and polarization coupling in the index-guiding photonic crystal fibers (PCFs) with squeezed hexagonal crystal lattice. The sign of form birefringence in such photonic crystal fibers can be changed as the wavelength changes, which is reported for the first time to our knowledge. A PCF-based polarization splitter with squeezed hexagonal lattice is proposed to achieve excellent extinction ratio and a simpler structure than those reported previously.
©2004 Optical Society of America
In recent years, research interests in photonic crystal fiber (PCF), which is featured by an array of air holes running parallel to the fiber axis, are increasing [1–2]. The PCFs are mainly divided into two classes: the photonic bandgap fiber and the air-silica index-guiding fiber. The latter utilizes the modified internal reflection principle [3–10] and forms cores by an absence of the air holes. The air-hole array acts as the fiber cladding with an average refractive index lower than the silica core. Thus the regularly distributed air-hole array is not necessary, which brings more possibilities to design new fiber structures. The index-guiding PCFs exhibit many unique properties of light guidance, such as single-mode operation over a wide range of wavelength , highly adjustable effective mode area and nonlinearity [4–6], the engineerable dispersion at visible and near-infrared wavelengths [7–10] and so on.
The PCFs can be made highly birefringent due to a sharp index contrast between air and silica. The photonic crystal fibers have exhibited their potential to realize high performance polarization-relative components including polarization maintaining fibers (PMFs) [11–14] and polarization splitters [15–17]. Some PMFs employ asymmetric air-hole distribution near the fiber core to produce a high birefringence [11–13]. They generally have a six-fold symmetric air-hole array used as the fiber cladding and thus the birefringence is localized in the core. For such PMFs, the mode field at the long wavelength extends far beyond the core region causing the birefringence to decrease. In Ref. 14 Steel et al discussed elliptic-hole photonic crystal fibers. The important distinction from other PMFs is the global birefringence over the core and the cladding of the fibers, which means that the birefringence increases as the wavelength becomes longer. However in all the reported PMFs, the fast and slow axes are determined by the fiber structures and do not swap as incidence wavelength is changed.
More recently, the PCF-based polarization splitters with identical dual cores are also proposed [15,16], in which strong polarization dependent coupling results in the different coupling lengths for the two polarization states. Another PCF polarization splitter working as a polarization selective coupler is demonstrated . Although these polarization splitters have good splitting performances and short lengths, they always employ some exaggeratedly enlarged and diminished air holes to produce the high birefringence at the expense of prohibitive fabrication difficulty.
Here we report a new type of the PCFs with squeezed hexagonal crystal lattice. This is another kind of the PCF-based PMFs exhibiting the global birefringence over the fiber core and cladding. One unusual polarization property of these fibers is that the fast and slow axes of the fibers can be reversed when one changes the incidence wavelength, which is revealed for the first time to our knowledge. We examine the birefringence as functions of air-hole diameter, lattice pitch and the incidence wavelength. Also, the squeezed crystal lattice has advantages in polarization splitting. A simple structure of the PCF-based polarization splitter is proposed, in which all air holes are set according to a moderate air-filling fraction to avoid the air hole collapse in the fiber drawing process. Thus the fabrication of the polarization splitter based on squeezed hexagonal PCF is feasible. And the excellent extinction ratio better than 18 dB is also achieved.
2. Polarization maintaining fibers
The new PCF is plotted in Fig. 1. All air holes are circular and anchored in a triangular lattice. The lattice pitches Λx and Λy in x- and y-axis directions are different. This structure is obtained as a result of squeezing the crystal lattice along x-axis direction, which introduces the form birefringence. The diameter d of all air holes is 0.62 µm, and Λx and Λy are 1.2 µm and 1.56 µm, respectively. The incidence wavelength is 1.55 µm. We employ the fully vectorial beam propagation method for modeling the propagation of the fundamental modes in such fibers . The x- and y-polarization states are individually launched into the fiber core, and the corresponding effective refractive indices are calculated using the algorithm proposed in Ref. 20. The numerical accuracy of the algorithm is in the order of 1e-6. The birefringence (neffy-neffx) of the fiber shown in Fig. 1 is -5.80×10-4.
We examine the influence of the air-hole size on the fiber birefringence. As shown in Fig. 2, the birefringence increases from -7.58×10-4 to 9.63×10-4 and changes its sign when the diameter of the air holes is enlarged from 0.39 µm to 0.86 µm. In particular, the birefringence almost equals zero when the diameter of the air holes is 0.78 µm. This might be understood as follows. Simulations indicate that the fast axis is y-axis in the fiber cladding, and for the elliptic core the fast axis lies in x-direction. The birefringences of the core and cladding regions have opposite signs. The total birefringence is determined by a combined effect of the core and the cladding. When the diameter of the air holes is large enough to limit almost all energy in the core, the core birefringence is dominant and the total birefringence is positive. In the case of small air holes, the confinement of the air holes on mode field becomes weak, and more energy flows out of the core and into the cladding region. The birefringence of the cladding overwhelms that of the core and determines the sign of the total birefringence. A non-birefringent turning point can be found in such a two-fold symmetric structure, which is attributed to a balance between the core birefringence and the cladding birefringence.
Figure 3 illustrates the birefringence as a function of the lattice pitch Λx in x-axis direction. The diameter of the air holes is fixed at 0.39 µm. The pitch Λx decreases from 1.2 µm to 0.7 µm. There is a significant variation of the birefringence from -7.58×10-4 to -8.217×10-3 as the pitch Λx is diminished. The decrement in the pitch Λx forces the x-polarization state to flow out of the core much more than the y-polarization state, which leads to a high birefringence and large polarization dependent losses. If the crystal lattice pitch is further diminished along x-axis, a large part of the energy is squeezed out of the core. For very small Λx, the holes may eventually overlap in the x direction, and the refractive index distribution tends toward a thin film stack with interfaces parallel to the x-axis [20,21]. The fiber loss can be effectively reduced by adding a lower index outer cladding with large air-filling-fraction hole array, which makes fiber equivalent to dual-cladding structure.
The wavelength dependence of the birefringence is given in Fig. 4. We set Λx=1.2 µm, Λy=1.56 µm, and the air-hole diameter is 0.78 µm. Wavelength varies from 0.53 µm to 2.13 µm, and the material dispersion has been considered in our simulations. The birefringence is firstly increased and then decreased through a zero-birefringence point. To our knowledge this is the first photonic structure having variable polarity with the incidence wavelength. A zero-birefringence wavelength is found close to 1.53 µm, and the birefringence at the 1.53-µm wavelength is 9×10-6. For the longer wavelength, more energy flows out of the core; the influence of the cladding birefringence is stronger than that of core ellipticity, which leads to negative birefringence. In the case of the short wavelength, the core birefringence dominates the cladding birefringence, which leads to positive birefringence. It is noticed that the effect of wavelength on the birefringence at the short wavelength is much smaller than that at the long wavelength,
The spacing of the air holes in the cladding, which is a kind of sub-wavelength structure, is smaller than the dimension in the core. The cladding birefringence is more sensitive to wavelength. At the long wavelength, the cladding birefringence is dominant, and thus wavelength dependence of the birefringence over the long wavelength region is more obvious. As for shorter wavelength, the birefringence becomes small once more because the air holes impose less confinement on the mode field and thus the contribution of the asymmetric air-hole arrangement to birefringence is weakened.
3. Polarization splitters
The discussions above indicate that the squeezed photonic crystal lattice can provide a large freedom to modify the fiber birefringence without deliberately increasing or diminishing the sizes of some air holes. This is important to design a polarization component with a simple air-hole array pattern. Here a PCF-based polarization splitter with the simplest structure so far is proposed.
Figure 5 shows the structure of the polarization splitter. The diameter of all air holes is 0.7 µm, and Λx and Λy are 1.2 µm and 1.56 µm respectively. The structure parameters are carefully designed to obtain a simple integer ratio of the coupling lengths of the two polarization states. Two air holes are lost to form the cores of the splitter. We note that the air holes are uniform and are set under a moderate air-filling fraction. The transition from a polarization maintaining fiber to a polarization splitter does not request the introduction of the enlarged air hole, which is beneficial to avoid any collapse of the air holes in the fiber drawing process.
45-degree linearly polarized light with 3-µm-FWHM Gaussian intensity distribution is launched into the core A. The incidence wavelength is 1.55 µm. A strong polarization dependent coupling is observed from the normalized power transferring curves plotted in Fig. 6. The two polarization states couple across the two cores at the obviously different speeds. The coupling lengths for x- and y-polarization states are 1.27 mm and 1.87 mm, and coupling length ratio is about 2:3. At the output of a 3.8-mm-long fiber the two polarization states are expectedly separated with the extinction ratios of 18.74 dB and 20.11 dB for x- and y-polarization states respectively. Intensity distributions of the mode fields at the splitter output are shown in Fig. 7. The narrow silica bridges between the two cores are utilized to achieve the strong polarization dependent coupling. The loss of the splitter is very low due to the short length of the fiber.
We have discussed the birefringence in the photonic crystal fibers with the squeezed hexagonal crystal lattice. The influences of the lattice parameters and wavelength on the birefringence in these fibers are examined. In such PCFs, a novel polarization property that the incidence wavelength is able to control the fiber polarity is demonstrated for the first time. We have also presented the polarization splitting with high extinction ratio by using the dual-core squeezed photonic crystal fibers.
The work is supported partly by the “Trans-Century Training Programme Foundation for the Talents by the Ministry of Education”.
References and links
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