In this paper the guiding properties of photonic crystal fibers with a square lattice of air-holes in a silica matrix have been studied for the first time. The dispersion curves of fibers with different hole-to-hole spacing and air-hole diameter have been accurately calculated. Negative values of the dispersion parameter and the dispersion slope have been obtained with a hole-to-hole spacing of 1 µm. A comparison between fibers with square and triangular lattice has been also performed, taking into account the dispersion properties and the effective area in the wavelength range between 1200 nm and 1600 nm.
©2004 Optical Society of America
Photonic crystal fibers (PCFs) have particular guiding properties, strictly related to the geometric characteristics of the air-holes in their cross-section, and have been successfully exploited in different application [1, 2, 3]. In particular, the dispersion properties of the triangular PCFs can be engineered by changing the hole-to-hole spacing and the air-hole diameter [4, 5, 6, 7]. It has been shown that triangular PCFs with a silica core can be succesfully used to compensate the positive dispersion parameter and dispersion slope of a single mode fiber  or can be designed to have a flat dispersion curve and the zero-dispersion wavelength around 1550 nm [6, 8, 9].
In this paper the guiding and the dispersion properties of PCFs with a square lattice of air-holes have been investigated for the first time. It is interesting to analyze how a regular air-hole disposition different from the triangular one, usually studied so far, can affect the characteristics of the guided mode. Moreover, it is important to understand in which terms all the results previously obtained for the triangular PCFs can be applied to the square-lattice ones. Recently, a square-lattice PCF preform has been realized with a standard fabrication process, stack and draw, in order to study the localisation and control of high frequency sound by introducing two solid defects in the periodic distribution of air-holes . Thus the technological feasibility of the square-lattice PCFs has been demonstrated, since the final PCFs can be obtained by drawing the intermediate prepared preforms .
The guiding and dispersion properties of square-lattice PCFs have been analyzed as a function of the geometric characteristics, that is the hole-to-hole spacing, or pitch Λ and the diameter d of the air-hole in the fiber cross-section (Fig. 1(a)). Simulation results have shown that the dispersion parameter values increase in the wavelength range considered, from 1200 nm to 1600 nm, for a fixed hole-to-hole spacing, chosen between 1 µm and 3 µm, as the air-holes become bigger. Moreover, it has been demonstrated that the square-lattice PCFs with the smallest pitch, that is 1 µm, have negative dispersion parameter in the wavelength range around 1550 nm. The square-lattice PCFs with small pitch and large air-hole diameter, whose dispersion slope is negative too, can be used, as done with the triangular ones , to compensate the positive dispersion and dispersion slope of the traditional single-mode fibers in the C band. A comparison has been made between the square-lattice PCFs and the triangular ones, taking into account the dispersion properties and the effective area of the guided mode. It has been shown that the square-lattice PCFs can have an effective area slightly larger than the triangular ones for all the considered wavelengths, due to the different air-hole position around the silica core.
2. Dispersion properties of the square-lattice PCFs
All the PCFs studied in this paper have a silica core, obtained by introducing a defect, that is by removing a hole, in the center of the fiber transverse section. Fig. 1(b) reports the first ring of air-holes of a square-lattice PCF and a triangular one with the same Λ and d values, showing a lower average value of the refractive index around the core in the triangular PCF. In fact, in this case the first ring comprises six air-holes whose distance from the core center is equal to Λ, thus resulting in a stronger field confinement.
The properties of the square-lattice PCFs have been accurately studied through a full-vectorial solver based on the Finite Element Method (FEM) [4, 11, 12]. The effective index neff for the different wavelengths in the range between 1200 nm and 1600 nm has been obtained by the FEM approach [4, 8]. Notice that the chromatic dispersion of silica has been taken into account through the Sellmeier equation [4, 8]. Starting from the neff values versus the wavelength, the dispersion parameter has been derived using simple finite difference formulas [4, 8]. In order to compare the guiding properties of the PCFs with different lattice, the effective area has been accurately evaluated applying the Poynting vector definition, starting form the computed mode distribution .
The influence of the geometric parameters Λ and d/Λ has been investigated considering PCFs with five rings of air-holes in the cross-section. For fibers with low Λ values, a FEM solver with Perfectly Matched Layers as boundary conditions which accounts for leakage losses has been used not to affect the dispersion results. Five values of the hole-to-hole spacing Λ, that is 1, 1.5, 2, 2.5 and 3 µm, have been chosen, and d/Λ has been varied in the range 0.5÷0.9.
Fig. 2 shows the dispersion parameter D(λ) of the square-lattice PCFs with different d/Λ values and Λ=1 µm, Λ=2 µm and Λ=3 µm, respectively, for the wavelengths between 1200 nm and 1600 nm. All the square-lattice PCFs with the smallest pitch, 1 µm, have negative dispersion parameter in the C band, around 1550 nm (Fig. 2(a)), since the core dimension is very little and the waveguide dispersion dominates on the material one [6, 7]. The minimum dispersion value at 1550 nm, -277 ps/km·nm, is obtained with the PCF characterized by Λ=1 µm and d/Λ=0.6. Notice that D values increase with the air-hole diameter, so only the PCFs with d/Λ≤0.7, that is with small air-holes, have negative dispersion parameter in all the wavelength range here considered. The fiber with the smallest air-holes, that is the one with d/Λ=0.5, has a dispersion curve with a minimum, about -248 ps/km·nm, around 1550 nm and a positive dispersion slope for the longer wavelengths. The other square-lattice PCFs, with d/Λ≥0.6, have negative dispersion slope, so they could be used as dispersion compensating fibers. In fact, as it has been already demonstrated for the triangular PCFs [6, 7], the influence of the waveguide dispersion decreases when the pitch becomes larger. This is confirmed also in Fig. 2(b) and 2(c) by increasing the pitch Λ to 2 or 3 µm, the dispersion parameter of all the PCFs becomes positive, independently from the air-hole dimension, that is from the d/Λ value. It is interesting to notice that, as the pitch increases, the dispersion slope of the curves becomes more positive. Moreover, a change of d/Λ value causes a smaller difference in the dispersion parameter values, of about 8 ps/km·nm, for the PCFs with the higher Λ, that is 3 µm. Notice that the dispersion curve of the square-lattice PCF with d/Λ=0.5 and Λ=2 µm is quite flat, around the value of 53 ps/km·nm, from 1425 nm to 1550 nm (Fig. 2(b)).
Fig. 3 allows to understand how the dispersion properties of the square-lattice PCFs change as a function of the pitch Λ for a fixed d/Λ value. In this case d/Λ=0.9 has been chosen, but the results are almost the same for different air-hole dimension. Notice that an increase of 0.5 µm in the pitch value, that is from 1 µm to 1.5 µm, causes a significant change in the dispersion curve. In fact, there is a great difference between the dispersion parameter values of the two PCFs, which increases with the wavelength, being about 56 ps/km·nm at 1250 nm and about 310 ps/km·nm at 1600 nm. Moreover, the dispersion slope, which is negative for the PCF with Λ=1 µm in all the wavelength range considered, becomes almost null, in the wavelength range between 1200 nm and 1450 nm, and positive, for the longer wavelengths for the PCF with Λ=1.5 µm. When Λ≥2 µm, the slope of the dispersion curves is always positive. Finally, the dispersion parameter values, which are all greater than 50 ps/km·nm for these PCFs, decrease as the pitch Λ increases from 2 µm to 3 µm.
3. Comparison between the square-lattice and the triangular PCFs
In order to make a comparison of the guiding and dispersion properties of PCFs with different geometric characteristics, a square-lattice PCF and a triangular one have been considered with five air-hole rings and the same values of Λ and d/Λ, that is 1 µm and 0.9, respectively. It has been already demonstrated that the triangular PCF with these geometric parameters has negative dispersion and dispersion slope, and can be successfully used as a dispersion compensating fiber for a NZDF . As it is shown in Fig. 4, both the fibers have negative dispersion, which is greater in module for the square-lattice PCF. For example, at 1550 nm D=-293 ps/km·nm for the triangular PCF and D=-157 ps/km·nm for the square-lattice one. The last PCF could better compensate the positive dispersion of a NZDF in a wide wavelength range, since its dispersion slope is lower around 1550 nm. In fact, its compensation ratio values at 1500 nm and 1600 nm, being 0.964 and 1.007 respectively, are closer to the optimum value, that is 1, than those for the triangular PCF, which are 0.913 at 1500 nm and 1.047 at 1600 nm . However, a square-lattice PCF longer than the triangular one would be necessary to completely compensate the dispersion of the NZDF at 1550 nm, due to its lower negative dispersion parameter value at this wavelength.
Moreover, the effective area values of the square-lattice PCFs are still small, being lower than 2 µm 2 in all the wavelength range considered, but they are higher than those of the triangular PCF. As an example, the square-lattice PCF has an effective area at 1550 nm which is 18% larger than that of the fiber with the triangular lattice. This difference can be explained by considering that the square lattice is characterized by a lower air-filling fraction f=(π/4)(d/Λ) 2, which is almost 86% of the one for the triangular lattice, that is f=(π/2√3)(d/Λ)2. As a consequence, the square-lattice PCFs provide higher values of the average refractive index of the cladding, that is a lower step index which results in a lower field confinement.
A final analysis on the properties of the square-lattice and triangular PCFs is reported in Fig. 5 for different values of the hole-to-hole spacing. A smaller d/Λ value, that is 0.5, has been chosen for the two fibers, so that the triangular PCF is singlemode in all the wavelength range considered also for the largest pitch Λ=3 µm [13, 14]. It is interesting to notice that the square-lattice PCF has higher dispersion parameter than the triangular one when the pitch is small, that is 1 µm, and lower D values when the hole-to-hole distance is large, that is Λ=3 µm (Fig. 5(a)). The dispersion slope is only slightly influenced by the geometric characteristics of the lattice, being similar for the two PCFs. The PCFs with the square lattice have larger effective area for both the pitch values here considered (Fig. 5(b)). In particular, there is a quite greater difference between the Ae f f values of the two kind of PCFs if the pitch is large, that is Λ=3 µm. Finally, notice the tight field confinement obtained in both the PCFs with d/Λ=0.5 and Λ=3 µm, due to the large core diameter (Fig. 6). Since the fundamental component of the magnetic field is all confined inside the first air-hole ring, its shape clearly underlines the differences in the position of the air-holes belonging to the inner ring and, as a consequence, the different geometric characteristics of the two lattices, that is the square and the triangular one. Notice that, due to their field shape, square-lattice PCFs could be useful if applied as pig-tail fibers of integrated optical devices with a rectangular or a square transverse section.
A thorough analysis has been performed for the first time on the guiding and the dispersion properties of a new kind of silica-core PCFs, characterized by a square-lattice of air-holes in the transverse section. Different values of the hole-to-hole spacing and of the air-hole diameter have been considered, in order to show how the PCFs properties are influenced by the geometric characteristics of the lattice. It has been demonstrated that the square-lattice PCFs with small pitch and large air-holes can be used as dispersion compensating fibers, due to their negative dispersion parameter and negative dispersion slope in a wide wavelength range around 1550 nm. Moreover, a comparison has been made between square-lattice and triangular PCFs with the same Λ and d/Λ, taking into account the dispersion parameter and the effective area values in the wavelength range between 1200 nm and 1600 nm.
References and links
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