1. Introduction

Light propagation in photonic crystal fiber (PCF) is due to either of two mechanisms: the index-guiding effect and the photonic bandgap (PBG) effect. In the former case the propagation of light is similar to that in totally internally reflected fibers and the field is trapped about a high-index defect. Conversely, in the latter case the average core index could be lower than the average cladding index and guidance is due to the PBG effect.

Much attention has been paid to air-core PBG fibers, which are formed by elimination of 7 or 19 air holes in the center [1, 2]. In Ref. 3 it was proposed that by downdoping of the central silica solid core, light can propagate in the defect core with the PBG effect. Confinement loss of this solid-core honeycomb PBG fiber was investigated in Ref. 4; simulations have shown that it is much easier to reduce the confinement loss in solid-core PBG fibers than in air-core PBG fibers.

Recently a square-lattice PCF preform was fabricated by a standard process [5], and the technological feasibility of a square-lattice PCF was demonstrated because the final PCF could be drawn from the intermediate perform. Chromatic dispersion D of the square index-guiding PCF was analyzed in Ref. 6, where the influence of lattice parameters (d and Λ, the air-hole diameter and the pitch, respectively) on the dispersion properties was demonstrated.

In this paper, a novel square solid-core PBG fiber is proposed. Downdoping the central silica of a squarelike PCF enables light to propagate in the defect core with the PBG effect. Simulations have shown that both the PBG’s width and the bandwidths of defect modes in square PBG fibers are greater than those in honeycomb PBG fibers. We analyzed the dispersion of this square PBG fiber thoroughly and found that dispersion of square solid-core PBG fibers is dominated by material dispersion and influenced by the PBG edges at the bandwidth boundaries, which are due to the confinement of light by the PBG effect.

2. Parameter optimization and simulation results

Square and triangular lattices are two kinds of typical PCF lattice. Replacing some air holes with silica periodically forms novel squarelike or triangularlike lattices, as shown in Fig. 1. The lattice in Fig. 1(b) is also called a honeycomb lattice. Doping the central silica parts of these two-kinds of lattice introduces solid-core defects (red circles in Fig. 1, with diameter Λ) into these two kinds of PCF, wherein light can propagate. If the central defects are downdoped, the effective indices (n _eff=β/k) of defect modes are less than the effective cladding index, and light can propagate with the PBG effect.

 

Fig. 1. Solid-core PBG fibers with (a) squarelike and (b) triangularlike lattices. The central red circle is the doped region; black and white areas are silica and air, respectively.

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The air-filling fractions of square and honeycomb PBG fibers in Fig. 1 are

$f_S=\frac{3\pi }{16}{\left(\frac{d}{\Lambda }\right)}^2\approx 0.59{\left(\frac{d}{\Lambda }\right)}^2,f_H=\frac{\pi }{{\left(\sqrt{2}+1\right)}^2}{\left(\frac{d}{\Lambda }\right)}^2\approx 0.54{\left(\frac{d}{\Lambda }\right)}^2,$,

respectively. For the same d/Λ, the relative variation of f_S and f_H is less than 10%.

In this paper the mode field pattern and the dispersion characteristics of a PCF are simulated by Hermite–Gaussian functions [7, 8], which provide an efficient model for analyzing the modes in PCF that takes only 1 min to accomplish by simulation on P4 2-GHz computer. We used this model to simulate the examples proposed in Ref. 3. Our results are consistent with those obtained by the authors of Refs. 7 and 8. Then we used this model to perform the simulations reported in this paper. We chose λ=1.55 µm, Λ=2 µm, and d/Λ=0.9; the modal field patterns of square and honeycomb PBG fibers are shown in Fig. 2. The contours of these two defect modes are square and hexagonal, respectively, as influenced by the primary lattice.

 

Fig. 2. Typical mode patterns in (a) square and (b) honeycomb solid-core PBG fibers with Λ=2 µm, d/Λ=0.9, and λ=1.55 µm. The units of the axes are micrometers.

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For both square and honeycomb PBG fibers we simulated n _eff of the defect modes with both updoped (for example, doped with germanium) and downdoped (for example, doped with fluorine) cores. Technically, the relative index variation (Δn) of doped silica can reach approximately 1–2%; in this paper the doping level is denoted Δn. The values of n _eff of these two kinds of defect mode are shown in Fig. 3; n _eff of the cladding mode [9] (red curve) and the fundamental PBG (blue and green curves for the upper and lower edges, respectively) are also plotted. The parameters chosen were d/Λ=0.9 and index of silica n=1.45.

 

Fig. 3. Effective indices (n_eff=Λ/k) for defect modes in (a) square and (b) honeycomb PBG fibers with d/Λ=0.90. The downdoped defect modes, from top to bottom, are for downdoping levels of 0.5%, 1%, and 1.5%.

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Figure 3 shows that, for both lattices, the n _eff of the updoped defect mode is higher than that of the cladding mode, which means that the index-guiding effect is the guiding mechanism of the updoped defect mode; in contrast, values of n _eff of downdoped defect modes are inside the PBG and are less than that of the cladding mode, which verifies that the PBG effect is the guiding mechanism of the downdoped defect modes. For the same parameters, both the PBG width and the bandwidth of the downdoped defect modes in square PBG fibers are greater than those in the honeycomb fibers. For example, at λ/Λ=1.3 the relative PBG widths in Figs. 3(a) and 3(b) are 10% and 8%, respectively; for the 1.5% downdoping level, the bandwidths of the defect modes in Figs. 3(a) and 3(b) are approximately 1.4 λ/Λ and 0.8 λ/Λ, respectively. Compared with the bandwidths of the defect modes in honeycomb PBG fiber, those in square PBG fiber are increased by ▭75%, whereas the variation of air-filling fraction is less than 10%.

Figure 4 shows the displacement fields of the Bloch waves [10] in square and honeycomb PBG fibers. We can see that the contrast of the displacement fields between low bands (or dielectric bands) and up bands (or air bands) in square PBG fiber is more than that in honeycomb fiber. Because the difference in the energy distribution of Bloch modes is responsible for the PBG, we can deduce that the PBG in square fiber is wider than that in honeycomb fiber.

 

Fig. 4. Intensity of displacement fields of Bloch waves of (a) a low band and (b) an up band in square PBG fiber and of (c) a low band and (d) an up band in honeycomb PBG fiber.

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The downdoped defect modes cut off at both short and long wavelengths because of the PBG effect, and different doping levels have different cutoff wavelengths. For the following dispersion simulations, λ was chosen between the two cutoff wavelengths to ensure PBG guiding. To investigate the dispersion properties of square solid-core PBG fibers thoroughly, in the following dispersion simulations we took into account material dispersion of silica through the Sellmeier equation.

First, the influence of downdoping levels on the dispersion characteristics of square PBG fiber was investigated. We set Λ=2 µm and d/Λ=0.9 and chose the downdoping levels of the defect core as 0.5%, 1%, and 1.5%. Dispersion of square PBG fibers for these three downdoping levels is shown in Fig. 5(a). We can see that D increases with λ and that the dispersion curves of different downdoping levels vary little, which indicates that the downdoping levels of the core have little influence on the dispersion of defect modes. The difference of dispersion at long λ is due to the PBG effect, for n _eff is close to the PBG upper edge (as shown in Fig. 3), which may influence the derivatives of n _eff (or dispersion).

Chromatic dispersion D is composed of waveguide dispersion and material dispersion. Because the claddings of square PBG fibers are the same, waveguide dispersion is identical for different doping levels. The trend that D increases with λ is dominated by material dispersion, which is the second-order derivative of the refractive index of doped silica:

$(D=-\frac{\lambda }{c}\frac{d^{2}n}{d{\lambda }^2}).$

If downdoping levels remain constant, for small Δn (▭1% in our simulation) the relative variation of D among various curves is of the same order as Δ; therefore, material dispersion is almost identical for different doping levels and the values of D of three curves vary little. To clarify what kind of dispersion dominates, we calculate the dispersion characteristic for the fiber, neglecting material dispersion (set n=1.45), which is compared with the case when material dispersion is considered, as shown in Fig. 5(b). The waveguide dispersion changes rapidly only at the boundaries of the bandwidth, as influenced by the PBG effect. Comparison of the two curves shows that dispersion characteristics in square solid-core PBG fiber are dominated by material dispersion and influenced by the PBG effect at the bandwidth edges.

 

Fig. 5. (a) Dispersion of square PBG fibers with Λ=2 m and d/Λ=0.90 for three downdoping levels: 0.5%, 1%, and 1.5%. (b) Comparison of D in square PBG fiber (downdoping level, 1%) of two cases: without material dispersion and with material dispersion.

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As the doping level has little influence on dispersion, for the following simulations the downdoping level was set at 1%. Then we investigated further the influence of cladding parameters (d and Λ) on the dispersion characteristics.

We fixed Λ and set d/Λ=0.9, 0.7, 0.5, respectively. Dispersions of Λ=1.5, 2, 3 µm are shown in Fig. 6. We simulated the dispersion in the whole region of wavelengths where wave guidance takes place; therefore, for different values of Λ, λ cut off at different wavelengths (λ/Λ near 0.0–1.6). In each figure, D is shown to increase with λ, and the three curves cross one another; this also indicates that material dispersion dominates waveguide dispersion in square PBG fibers, which are different from the index-guiding square PCFs described in Ref. 6. In PBG fiber, light is confined in the core by the PBG effect, the cladding structure parameters (d and Λ) have less influence on the propagation of light in the core, and dispersion is determined mainly by the characteristics of the core.

 

Fig. 6. Dispersion of square solid-core PBG fibers for d/Λ=0.9, 0.7, 0.5. (a) Λ=1.5 µm, (b) Λ=2 µm, (c) Λ=3 µm.

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Finally, we fixed d/Λ=0.9 and chose Λ=1.5, 2, 3, 4, 5 µm. Dispersion of square solid-core PBG fibers is shown in Fig. 7. D decreases with increasing Λ. In the majority range of λ, dispersion changes gently, but, at the longer side for Λ=1.5 µm and at the shorter side for Λ=5 µm, D changes steeply because of the PBG effect. From Fig. 3(a) we can see that for Λ=1.5 µm, when λ>1.5 µm and λ/Λ>1, n _eff of the defect mode is close to the upper PBG edge; for Λ=5 µm, when λ<1.2 µm and λ/Λ<0.24, n _eff of the defect mode is close to the lower PBG edge. n _eff of the defect mode is distorted because of the PBG effect close to the PBG edges, which may cause large dispersion and dispersion slope in these regions.

 

Fig. 7. Dispersion of square solid-core PBG fibers for several values of Λ with d/Λ=0.90.

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Zero-dispersion wavelength λ₀ of the defect mode increases with Λ, and one can easily tune λ₀ to the desired wavelength by adjusting the geometric parameters. Figure 6 shows that, when Λ increases from 1.5 to 3 µm, λ₀ shifts from the 0.6- to the 1.0-µm region, and the effective modal field areas A _eff for these two cases are ▭2 and ▭7 µm², respectively; therefore these square solid-core PBG fibers can be used in some nonlinear applications at short wavelengths. If the solid core is doped with Er³⁺ or other rare-earth element ions in addition to fluorine, interaction between the light emitted from these ions and the PBG effect in the cladding may lead to light emission with high efficiency, narrow spectral linewidth, and improved directionality, which may be used to fabricate fiber lasers with ultralow thresholds or to reduce the amplified spontaneous emission of rare-earth-doped fibers.

3. Conclusions

A novel solid-core PBG fiber with a square lattice has been proposed in this paper. Simulations have shown that light can propagate in a downdoped defect core by the PBG effect. With same PCF parameters (d and Λ), the PBG width and the bandwidths of defect modes in square solid-core PBG fibers are both greater than in honeycomb solid-core PBG fibers. Dispersion of the defect modes in square solid-core PBG fibers has been thoroughly analyzed. Simulations have shown that the dispersion of solid-core PBG fiber is dominated by material dispersion and influenced by the PBG effect at the bandwidth boundaries.