In this work, modal birefringence and polarization-dependent leakage loss for the fundamental mode in elliptical-hole rectangular-lattice photonic crystal fibers (EH-RL-PCFs) with modified air holes near the core are investigated by using a full-vector, finite-difference method in the frequency domain. Numerical results show that by introducing modified air holes, a heightened modal birefringence of the order of 10-2 has been obtained within the wavelengths ranging from 1.0 to 1.6 μm in the proposed PCFs, with a lattice length of 2 μm and with a large lattice length of 3 μm at 1.55 μm.
©2007 Optical Society of America
The development of new fiber designs is immediately impacted by the introduction of new concepts of guidance and cross-section in optical fibers. Fibers with different guidance concepts and periodic structures have been widely investigated during the past decade in several areas such as total internal reflection photonic crystal fibers (index-guiding PCFs), photonic band gap PCFs, Bragg fibers, and cylindrically symmetrical hollow fibers [1–4]. The above fibers result in a wide variety of appealing features that clearly surpass those of traditional fibers such as endlessly single mode , broadband negative chromatic dispersion, negative-dispersion slope, and high birefringence fibers .
In many practical applications, highly birefringent polarization-maintaining fibers (PMFs) are either preferred or required. To our knowledge, there are three methods of producing high birefringence in index-guiding PCFs. One method uses the application of stress to the core as in conventional PMFs, such as in PANDA and bow-tie fibers; the second method is realized by introducing asymmetry near the defect core of the PCFs ; and the third method is by designing an air-hole lattice or a microstructure lattice with inherent anisotropic properties, such as the elliptical-hole PCFs , liquid-crystal PCFs , rectangular-lattice PCFs , and squeezed hexagonal-lattice PCFs . The birefringence of these highly birefringent polarization-maintaining PCFs (HB-PM-PCFs) in the last two methods (10-3 or 10-2) is always 1 or 2 orders of magnitude higher than that of the conventional stress-induced birefringence fibers (10-4). By introducing a noncircular defect core, photonic bandgap PCFs with high birefringence can be achieved experimentally also .
In order to obtain HB-PM-PCFs with higher birefringence, the above methods have been combined according to recent literature. EH-RL-PCFs that have been studied owe the induction of highly increased birefringence to both elliptical holes and rectangular lattices [11, 12]. The studies of both elliptical-hole, hexagonal-lattice PCFs with two big air holes adjacent to the core and elliptical-hole, squeezed hexagonal-lattice PCFs were reported in [9,13] also. For even higher birefringence, a novel HB-PM-PCF is proposed in this work. This PCF is composed of rectangular lattices, elliptical air holes, and two modified air holes adjacent to the solid silica core. This structure can gather these three factors leading to highly increased birefringence directly. In our calculation the birefringence in such a HB-PM-PCF can reach 10-2 over the wavelength span from 1.0 to 1.6 μm and can be up to 3.83×10-2 at 1.55 μm, which is the highest level to date, to our knowledge.
2. Birefringence and leakage loss in EH-RL-PCFs with modified air holes
The general form of the proposed HB-PM-PCFs is illustrated in Fig. 1. The PCFs are characterized by the area S * cl of one elliptical air hole in the cladding, the lattice length Λ, the lattice width W, and the ellipticity ratio η = b/a, where b and a are the lengths of the major and minor axes, respectively. The area of one modified air hole near the core is S * co, and the ellipticity ratio η co equals b co/a co, where b co and a co are the lengths of the corresponding major and minor axes. The long side of the rectangular lattice and the major axis are directed along the y axis, while the short side and minor axis are parallel to the x axis. The width and height ratio of the lattice γ the dimensionless areas S cl and S co, are also defined as γ= w/Λ, S cl = S cl/Λ 2 and S co = S * co/Λ 2 , respectively. There are five air-hole rings in the calculations.
To analyze modal birefringence and polarization-dependent leakage loss properties of the proposed HB-PM-PCF, a full-vector finite-difference method in the frequency domain  is applied. Because of the geometric symmetry of the PCF, only one-quarter of the structure needs to be considered by applying a combination of PEC (perfect electric conductor) and PMC (perfect magnetic conductor) [Fig. 1(b)]. The refractive index of the air holes is set to 1 in the computation, and the background refractive index is obtained from the Sellmeier equation for silica glass .
Silica glass material is assumed to extend to infinity uniformly, and PML layers are used outside the computation region. The modal birefringence and the polarization-dependent leakage loss  can be determined according to the following formulations:
where β x and β y are the propagation constants of the orthogonal polarization modes, λ is the wavelength of the light, and Im[neff] is the imaginary part of the effective refractive index.
The birefringence of EH-RL-PCFs without modified air holes has been studied with various η and γ , while the polarization-dependent leakage loss and birefringence of the PCFs with different γ have also been investigated . Accordingly, fixed S cl = 0.04π, η = 4, and γ= 0.4 are chosen for our simulation. Only the value of S co is changed to study the influences on the leakage loss and birefringence, which are determined by b co and a co. For the purpose of the practical PCF fabrication, Λ is assumed to be 2 μm and 3 μm for comparison.
In EH-RL-PCFs, the cut-off frequency of the second-order mode is defined as the lowest frequency at which the second mode emerges and is usually very high, meaning an extensive region of the single mode in the PCFs. Even though there are higher-order guided modes existing in the PCFs at a short wavelength, the birefringence becomes very weak, and the leakage loss of higher-order modes is much larger than that of the two fundamental modes. Here we only take the polarization-dependent leakage loss and birefringence of the fundamental modes into account.
Figures 2 and 3 show the birefringence of fibers with fixed a co and b co respectively. According to the figures, the modal birefringence increases over the whole wavelength range from 1 to 1.6 μm, and the same S co structures with smaller Λ exhibit higher birefringence. The latter case can also be drawn in other HB-PM-PCFs [11, 12]. The birefringence is of the order of 10-2 for Λ = 2 μm over a long wavelength span, and even for Λ = 3 μm the birefringence at 1.55 μm is almost 1×10-2, which means much more facility in the practical fabrication of HB-PM-PCFs. Meanwhile, for fixed Λ and length of either axis (a co or b co), the larger the S co is, the higher the birefringence is. This is because a larger S co will induce the defect core of the PCFs to be more asymmetrical.
In order to further illuminate the effect of modified air holes on birefringence in EH-RL-PCFs, we defined a birefringence increment ratio (BIR) as follows:
where B ref (λ) is the birefringence of EH-RL-PCFs without modified air holes. As shown in Fig. 4, with S co = 0.075π, 0.0675π, 0.06π, a co= 0.3Λ, and Λ = 2μm, the BIR is above 9% for the three curves, especially beyond 60% with S co of 0.075π. It clearly supports that introducing modified air holes near the core in EH-RL-PCFs is an effective method for designing HB-PM-PCFs. It also can be found that the three curves decrease slowly when the wavelength increases. We attribute this phenomenon to the fact that the birefringence is much lower at a short wavelength than that at a long one, and a little increment of birefringence makes the BIR larger at a short wavelength.
To examine the availability of the proposed PCFs in practical applications, Fig. 5 and Fig. 6 show the leakage loss of the PCFs for x- and y-polarized modes with Λ=3μm, b co=Λ, and Λ=2μm, b co=Λ, respectively. It is apparent from these two figures that the loss of the x-polarized mode is greater than that of the y-polarized mode, especially at the larger S co. This can be observed in elliptical-hole, squeezed hexagonal-lattice PCFs  with the ellipses oriented along the y-direction, but it is the opposite in elliptical-hole, square-lattice PCFs (γ= 1)  with the same orientation of ellipses. In the latter PCFs, as a smaller a of elliptical holes in the x-axis between two neighboring air holes implies a lower air filling ratio, the y-polarized field leaks more to the cladding than the x-polarized field does. But in EH-RL-PCFs (γ> 1), owing to the squeezed lattice along the x-direction, the width of silica glass in the x-axis between two neighboring air holes decreases, but nothing is changed in the y-axis. If we keep squeezing the lattice, the loss ratio of the y-polarized mode to the x-polarized mode decreases and is less than 1 for certain γ. In Fig. 5 and Fig. 6, the loss difference between the two polarized modes increases with increments of S co but is too small to achieve the single polarization transmission in the PCFs. It is also shown that structures at Λ=3 μm exhibit very low leakage loss and the loss increases quickly at Λ=2μm, which induces high birefringence as illustrated in Fig. 3. There is a trade-off between the high birefringence and the low loss. In practice it should not be a problem, as loss can be significantly decreased just by adding airhole layers. So the proposed PCFs can be used as highly birefringent, polarization-maintaining fiber with low polarization-dependent loss.
The field patterns of x- and y-polarized fundamental modes at the wavelength of 1.55 μm with Λ=2 μm in Fig. 7 are for two different structures with S co = S cl = 0.04π, b co= b cl=0.8Λ, and S co = 0.075π, b co= Λ. Both of the orthogonal modes are well-confined in the core for each case, and the field patterns in the PCF with modified air holes are narrower than those in the unchanged one. This is because fields are pushed more to the core by introducing modified air holes, resulting in leakage from the core domain.
For the sake of achieving higher birefringence, the value of γ is reduced. The calculated birefringence of the proposed PCFs with γ=0.24 as a function of the wavelength is plotted in Fig. 8. The curves also increase monotonically, with the wavelength increasing as shown in Fig. 2 and Fig. 3. In spite of the same Λ that is adopted, the curves in Fig. 8 nearly doubled the value of the birefringence over the whole wavelength range compared with Fig. 3. The birefringence at the wavelength of 1.55 μm surprisingly increases to 3.83×10-2 for the structure with modified air holes and is enlarged distinctly in contrast to that of the PCF without changes.
The above studies are carried out theoretically, but in practical fabrication there may be various issues for the proposed PCFs. The deformation of the air-hole pattern due to surface tension, especially for elliptical holes, cannot be avoided and sometimes can affect the results severely. To appraise the effect of this imperfection, the lengths of the major axes of all air holes are multiplied by a pseudorandom value, respectively. These values are from a matrix in which elements are corresponding to air holes in the PCF one by one and are drawn from a normal distribution with a mean of 1 and a standard deviation of 0.08. This operation is also executed on the lengths of the minor axis but with a different pseudorandom matrix. We recalculate the birefringence and plot it in Fig. 8.
Compared to the ideal structure, the birefringence in the PCFs with random operation decreases a little, but the trend of the curves is invariable. Furthermore, the bending loss in the PCFs is evaluated by using the equivalent index model . As shown in the inset of Fig. 8, the bending loss of the fibers with different parameters of S co = 0.05π, b co= Λ and Λ = 2μm and S co = 0.05π, b co=Λ and Λ = 3μm, is quite low and of the same level of confinement loss. It changes slowly when the bending radius increases and indicates its usefulness in practice. PCFs with elliptical holes have been experimentally realized  and the method of stacking a rectangular lattice PCF has been shown , which is just like that of ordinary PCFs. So it is possible to draw EH-RL-PCFs with modified air holes with new fabrication methods continually emerging.
In conclusion, we have illustrated in detail the effects of structure parameters on the birefringence and polarization-dependent leakage loss of EH-RL-PCFs with modified air holes near the core. The extraordinarily high birefringence of 3.83×10-2 at 1.55 μm is achieved in the proposed PCF, which should be the highest value in publications to date. Even with the large Λ=3 μm, the birefringence at 1.55 μm is almost up to 1 × 10-2. The PCFs may be useful as HB-PM-PCFs in practical applications.
This work was supported by NSFC under grants 60477033 and 60671006. D.X. Yang may be reached via e-mail at firstname.lastname@example.org cn.
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