## Abstract

We propose a modified rectangular lattice PCF and numerically investigate its birefringence and dispersion. Based on the plane wave expansion method, it is shown that the proposed structure provides high birefringence and negative dispersion. Numerical results show that the birefringence of the modified PCF reaches 10^{-2} and the leakage loss is about 1000 times smaller than that of an original rectangular PCF because the modification gives rise to the strong confinement of guided modes. Dispersion and its slope are also negative over the C band.

©2009 Optical Society of America

## 1. Introduction

Photonic crystal fibers (PCFs) consisting of a periodic distribution of air holes along its length and a defect region in its center, have been intensively studied in recent years due to their unique optical properties [1–7]. In conventional PCFs, the core-guidance of the optical signal is provided by a solid silica defect core surrounded by periodic air-hole arrays in the cladding. Since the core refractive index is higher than the effective cladding index, which is an average of air holes and background silica, light signal can be guided by the total internal reflection along the silica defect core similar to conventional optical fibers.

High level of birefringence in fiber optics is required to maintain the linear polarization state by reducing polarization coupling. Due to the large index contrast of PCF compared to the conventional fiber, highly birefringent (HB) PCFs have been reported by breaking the circular symmetry implementing asymmetric defect structures such as dissimilar air hole diameters along the two orthogonal axes [5–6], asymmetric core design [7–8] and designing an air hole lattice or a microstructure lattice with inherent anisotropic properties such as the elliptical-hole PCF [9–10], and squeezed hexagonal-lattice PCFs [11–12]. Modal birefringence in these HB PCFs has been predicted to have values an order magnitude of 10^{-3} or 10^{-2} higher than that of the conventional HB fibers (10^{-4}) [13]. Another kind of highly birefringent structure reported is a rectangular lattice PCF [14–17]. According to the symmetry theory, the rectangular lattice is potentially more anisotropic than the triangular and honeycomb lattices [14]. The values of birefringence for basic rectangular lattice PCF reported in [14] have an order of 10^{-3}. By combining the elliptical-hole and rectangular lattice, the birefringence increases to an order of 10^{-2} [15–17]. However, elliptical air holes are very difficult to control during the fabrication process. In addition rectangular lattice PCFs, a deformed supersquare lattice PCF with high birefringence is reported [18].

Also, the dispersion properties of square lattice PCFs have been reported by Bouk et al [19]. In the study, it has been demonstrated that the square lattice PCFs with the smallest pitch, that is 1 μm, have negative dispersion in the wavelength range around 1.55 μm. The square lattice PCFs with small pitch and large air hole diameter, whose dispersion slope is also negative, can be used to compensate the positive dispersion and dispersion slope of the traditional single-mode fibers in the C band.

In this paper, we propose a modified rectangular lattice PCF which exhibits high birefringence, extremely low leakage loss, and dispersion compensating property in the C band simultaneously. Modal birefringence, chromatic dispersion and the leakage loss have been numerically analyzed by plane wave expansion method [20–21]. Numerical results show that birefringence of the modified rectangular lattice PCF can reach 10^{-2} and the leakage loss can be about a few thousand times smaller than that of a conventional rectangular PCF because the modification gives rise to the strong confinement of guided modes. Dispersion and its slope can also be negative over the C band.

## 2. Model of a modified rectangular lattice PCF

The conventional and modified rectangular lattice PCFs are shown in Fig. 1. The geometry of a modified rectangular lattice PCF looks like to a simple square lattice PCF with two-basis of air holes. The modified structure is proposed as shown in Fig. 1(b) by introducing another air hole between two air holes along x-axis for every other line to a conventional rectangular lattice PCF (Fig. 1(a)). The proposed modification gives merits in the following viewpoints.

In order to improve the capability of the dispersion compensation in the rectangular lattice PCF, another air hole between two air holes along *x*-axis for every other line are introduced to the conventional rectangular lattice PCF, which is called the modified rectangular lattice PCF. It is explained that the added air holes make the distance between air holes along the *x*-axis shorter and it enhances the dispersion compensation.

On the other hand, by introducing the added air holes, the index contrast between core and cladding gets higher than that of the conventional rectangular lattice PCF. Therefore, the higher birefringence can be achieved compared to that of the conventional rectangular lattice PCF, while maintaining the improved dispersion compensation.

We employed the plane wave expansion method to study optical properties of rectangular PCFs. The calculation accuracy in the plane wave expansion method is influenced by several parameters such as the number of plane waves, the size and shape of a supercell, and the tolerance [20]. By a lot of trial and error in the calculation for the perfectly symmetric PCF while varying these computational parameters, the optimum parameters are chosen based on the_convergence of the mode indices and the computation time. In our calculations, the optimum parameters were chosen such as; a supercell size of 6.5Λ×11Λ as a rectangular shape, the tolerance of 10^{-7} and the resolution of 256×256 (grid numbers). The error due to the finite number of the plane waves was to be less than 5% [20–21].

## 3. Optical properties of the modified rectangular lattice PCF

Figure 2 shows the field distributions of the fundamental modes with *λ*= 1.55μm in the conventional and modified rectangular lattice PCFs when Λ = 2.0 μm, *d* / Λ = 0.4. One can see clearly that the mode confinement of the modified rectangular lattice PCF is stronger than that of the conventional rectangular lattice PCF. The stronger mode confinement can give rise to higher birefringence and lower leakage loss.

The modal birefringence, the chromatic dispersion and the leakage loss [17] can be determined according to the following formulations:

The modal birefringence of the conventional and modified rectangular lattice PCF with Λ=2.0 μm and *d* / Λ=0.4 is shown in Fig. 3(a). From Fig. 2, by modification of rectangular lattice PCF, the mode confinement to the core is enhanced and modal birefringence increases to an order of 10^{-2} at 1.55 μm. In addition, the slope of birefringence in the modified rectangular lattice PCF is steeper than that of the conventional rectangular lattice PCF. At longer wavelength, therefore, the difference of birefringence for these two types of rectangular lattice PCF is much larger.

In order to demonstrate the enhancement of mode confinement in the modified rectangular lattice PCF, the leakage loss of the conventional and modified rectangular lattice PCF with Λ=2.0 and *d* / Λ=0.4 is calculated. From Fig. 3(b) the leakage loss of the modified rectangular lattice PCF have been significantly reduced when compared to the conventional rectangular lattice PCF from 10^{2} dB/m to 10^{-2} dB/m.

Chromatic dispersion is one of the most important properties of the PCFs. The dispersion properties of square lattice PCFs have been investigated [19]. From the study, the square lattice PCF with smallest distance between air holes, that is 1 μm, have negative dispersion parameter and negative dispersion slope in the wavelength range around 1.55 μm. To demonstrate above properties, the chromatic dispersion of the basic and modified rectangular lattice PCF with Λ=2.0 μm and *d* / Λ=0.4 is shown in Fig. 3(c). For the modified rectangular lattice PCF, the distance between nearest air holes is 1 μm. The dispersion compensating properties of the modified rectangular lattice PCF is significantly improved.

It is natural that the modified rectangular lattice PCF (Fig. 1(b)) shows higher birefringence and lower leakage loss than the conventional rectangular lattice PCF (Fig. 1(a)). This is because the modified rectangular lattice PCF has the more air holes in the cladding. However, the added air holes for every other line in the modified rectangular lattice are required not only to control the property of dispersion compensation but also to increase the index contrast with maintaining the single mode operation. In order to obtain the higher index contrast, if the conventional square lattice with the same x- and y- period with the core formed by three missing air holes in a row is suggested, single mode operation does not work any more in this more condensed structure. The specific advantage in introducing holes only for every other line in the modified rectangular lattice PCF is that three crucial properties of optical fiber such as high birefringence, dispersion compensation and single mode operation is possible.

## 4. Influence of the geometric parameters on birefringence and dispersion

In this section, we investigate the influence of the geometric parameters Λ and *d* / Λ on the birefringence for a modified rectangular lattice PCFs, which are illustrated in Fig. 4(a) and Fig. 4(b), respectively. Fig. 4(a) shows the birefringence value by varying Λ from 1. 6 μm to 2.4 μm when *d* / Λ = 0.4. The birefringence decreases with the increase of Λ. Fig. 4(b) shows the birefringence value by varying *d* / Λ from 0.2 to 0.4 when Λ = 2.0 μm. The birefringence increases with the increase of *d* / Λ. Therefore, in order to obtain high birefringence with an order of 10^{-2}, the PCF should be characterized by Λ < 2.1μm and *d* / Λ > 0.38.

The influence of the geometric parameters Λ and *d* / Λ on the dispersion properties of a modified rectangular PCF has been also investigated. Fig 5 shows the dispersion parameter *D* (*λ*) of the modified rectangular lattice PCFs with different *d* / Λ values and Λ = 1.6 μm, Λ = 2.0 μm, and Λ = 2.4 μm, respectively, for the wavelengths between 1.0 μm to 1.8 μm. *d* / Λ has been varied in the range from 0.2 to 0.4. For the modified rectangular PCFs with Λ = 1.6μm, the minimum dispersion value at 1.55 mm, -275 *ps* / *km.nm* is obtained with the PCF characterized by Λ = 1.6μm, and *d* / Λ =0.35 (Fig. 5(a)). As *d* / Λ varies from 0.4 to 0.2, the dispersion slope changes from -0.49 *ps* / *km.nm*
^{2} to 0.15 *ps* / *km.nm*
^{2}. For a modified rectangular lattice PCFs with *d* / Λ > 0.3, have negative dispersion value and dispersion slope in the C band. This has been already demonstrated for the square lattice PCF [19]. This property is also confirmed in Fig. 5(b) and Fig. 5(c) for a modified rectangular lattice PCF with Λ = 2.0 μm and Λ = 2.4 μm. As Λ increases, the dispersion parameter increases and the dispersion slope becomes gentle. For a modified rectangular lattice PCF with Λ = 2.4 μm and *d* / Λ = 0.4, the dispersion slope at 1.55 μm is negative, -0.14 *ps* / *km.nm*
^{2}. However the dispersion parameter is 40 *ps* / *km.nm.* In order to use a modified rectangular lattice PCF as a dispersion compensating fibers, the PCF should be characterized by Λ < 2 μm and *d* / Λ > 0.3.

As the Hi-Bi PCF is widely used in fiber loop mirrors as a major component for optical fiber sensing applications [22, 23], the added property of negative dispersion would provide better performance or another degree of freedom for the fiber sensor design. Also, a high birefringence (Hi-Bi) optical fiber may be very useful in a polarization maintaining transmission system if it has large negative dispersion. As the Hi-Bi PCF with negative dispersion is not a distributed fiber with negative dispersion, it is well suited as a functional block for the transmission system [24]. Therefore, the proposed modified rectangular lattice PCF will find applications in optical fiber sensing and polarization maintaining transmission system.

The fabrication technique of a rectangular lattice PCF with circular air holes is explained in detail in Ref. [14] using the stack and draw procedure, just as for the conventional PCFs. The fabrication of PCFs with circular air holes is generally easier. Therefore, without the complex fabrication method such as sol-gel method [25] and perform drilling [26], the proposed structure is believed to be attainable with a high feasibility.

## 5. Conclusion

We have proposed a modified rectangular lattice PCF which can be used as a dispersion compensating fiber while maintaining its input polarization states and single mode operation, and investigated its polarization and dispersion properties based on the plane wave expansion method. With the design, the high birefringence to the order of 10^{-2} was reached by optimizing Λ and *d* / Λ. The leakage loss could also be reduced significantly. To be stressed, the modified rectangular lattice PCF with Λ < 2 μm and *d* / Λ > 0.3 showed the negative dispersion value and negative dispersion slope, which is indispensable for the dispersion compensating fiber. Therefore, it is expected that the proposed modified rectangular lattice PCF with small Λ and large *d* / Λ can be used as high birefringence and dispersion compensating fibers over the C band.

## Acknowledgments

This work was supported by the Ministry of Education, Science and Technology of Korea through the APRI-Research Program of GIST and Center of Subwavelength Optics (R11-2008-095-01000-0).

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