1. Introduction

The introduction of new concepts of guidance in optical fibers has had an immediate impact in the development of new fiber designs. Specifically, fibers that show photonic band gaps present unusual guiding properties with high technological interest. Two main classes of these fibers have recently emerged. On the one hand, we have the fibers based on two-dimensional (2D) photonic crystals, the so-called photonic crystal fibers (PCF’s) [1, 2]. The transverse section of a PCF is a 2D silica-air photonic crystal with an irregularity of the refractive index, thus generating a defect in the otherwise regular structure. Guidance occurs in the region where the defect is located and is attributed to the inhibition of transverse radiation produced by the photonic-crystal cladding. As the second class we consider the Bragg fibers [3, 4], that are formed by a low-index core surrounded by an alternating cladding of high- and low-refractive-index layers. The light is confined to the core by cylindrical Bragg reflection off the alternating layers. One variant of the latest fibers is the coaxial Bragg fiber [5, 6], also named OmniGuide fiber, which tries to emulate the properties of conventional coaxial cables for optical frequencies. In this case, the cladding reproduces the behavior of an omnidirectional dielectric mirror. In the above context, we would like also to mention the cylindrically symmetrical hollow fiber [7, 8], a kind of PCF where the air holes lie on circular rings, that can be considered as an admixture of the two previous classes of band-gap structures.

The presence of a periodic, or almost periodic, structure as a cladding in such microstructured fibers determines the existence of photonic band gaps in the visible or in the near infrared region for some specific configurations. The dispersion relations of the guided modes lie on the forbidden (band gap) regions and, of course, are affected by the microstructure of the cladding.

The above photonic-band-gap guiding principle results in a wide variety of appealing features that clearly surpass that of traditional step-refractive-index fibers. In this paper we focus our attention on the group-velocity-dispersion properties of a particular type of these structures. The characteristics of the dispersion relations of the guided modes in fibers using photonic band gaps are explicitly reflected on the qualitatively different behavior of the chromatic dispersion, D. In this way, a suitable design of the geometric parameters in PCF’s leads to nearly-zero ultraflattened dispersion [9–11] or results in new tunable flattened dispersion properties [12]. Likewise, a zero-dispersion wavelength [13, 14] or very large negative dispersion values [15] can be achieved in single-mode cylindrically symmetric layered fibers.

Our interest in this work is to propose a new cylindrical multilayered fiber that can be regarded as a high-index-core Bragg fiber. It is a cylindrically symmetric microstructured fiber having a high-index core (silica in our case) surrounded by a multilayered cladding made of alternating layers of silica and a lower refractive-index dielectric. In contrast to standard Bragg fibers, guidance is not given in the low-index core, but in the silica core like in central-hole-missing PCF’s. Because the dispersion produced by the fiber geometry exists in all cases (even guiding light in air), the aim of our scheme is to compensate for the geometric fiber dispersion by means of the material dispersion corresponding to the material where the mode is propagating (silica in our case). Accordingly, using the remarkable dispersion properties due to the Bragg fiber geometry, we can properly compensate the material dispersion of the silica and, then, to achieve a very nice control of the dispersion behavior.

 

Fig. 1. Schematic diagram of a high-index-core Bragg fiber.

Download Full Size | PDF

2. The high-index-core Bragg fiber

The multilayered structure geometry is characterized by the radial multilayer period, Λ, and the low-index-layer thickness, a, as illustrated in Fig. 1. The refractive indices are alternately n _I and n _II, where n _II<n _I. In fact, we have selected the refractive indices of silica and air as n _I and n _II, respectively. Of course, our treatment can be easily adapted to other materials. Needless to say that, in order to provide structural support to the fiber, air should be replaced by some dielectric with a low refractive index, but the main results of this work would still apply. We have simulated different fiber configurations. For simplicity we restrict ourselves to vary only two independent structural parameters that define the Bragg geometry, Λ and a. Note that with this selection the core radius coincides with the thickness of the high index layer of the cladding. In any case, the structure is, strictly speaking, not periodic.

Our numerical calculations are performed with a 2D full-vector modal method developed by our own group that permits to incorporate the material dispersion in a natural way [16], combined with shield boundary conditions. According to general waveguide theory, guided modes, and also radiation modes, in a circularly symmetric waveguide can be labeled according to their azimuthal or angular symmetry order [17]. We have taken advantage of this fact to solve the different angular orders, ν, separately. In practical terms, the use of rotational symmetry reduces the original 2D problem (fields depending on the two transverse coordinates) into a 1D one (fields depending on the radial coordinate only). This provides not only a higher accuracy for a given number of auxiliary modes, but also the different band structures for every angular order.

Figure 2 shows the two different band-gap structures and the modal dispersion curves for the guided modes in a silica-air, high-index-core Bragg fiber for ν=1 and ν=0, including in the latter case the spectrum of TE modes and omitting TM modes as they are very similar. As in the original Bragg fiber, guided modes appear in the forbidden band gaps at their corresponding angular order ν. We point out that this structure exhibits guided modes simultaneously above and below the first conduction band, as in PCF’s [2]. In particular, the single guided mode shown in the upper forbidden band for ν=1 is the fundamental mode of the Bragg fiber, which, due to the rotational symmetry of the index profile, corresponds to a polarization doublet HE₁₁, as denoted in standard waveguide theory. No other higher-order modes of any angular sector appear in this band for the selection of the geometric parameters fixed in Fig. 2. The transverse intensity distribution for any of the two polarizations of the fundamental mode, HE₁₁, and the first intraband guided mode, TE₀₁, in Fig. 2 are visualized in Figs. 3(a) and 3(b), respectively, for λ=0.8 µm.

 

Fig. 2. Band-gap structure and modal dispersion relation curves for two angular sectors: (a) ν=1 (HE modes), and (b) ν=0 (only TE modes). In both cases, Λ=1.190 µm and a=0.248 µm. Conduction bands are represented by the shaded regions.

Download Full Size | PDF

 

Fig. 3. Transverse intensity distribution for: (a) the fundamental guided mode HE₁₁ in Fig. 2, and (b) the first intraband guided mode TE₀₁ in Fig. 2. In both cases, λ=0.8 µm.

Download Full Size | PDF

3. Dispersion results

We have extensively studied the dispersion properties of different high-index-core Bragg-fiber designs. We expect to tailor the chromatic dispersion of such fibers by manipulating the geometry of the multilayered cladding. The total dispersion, D, can be given, as a first approximation, in terms of the material dispersion, D_m, and the geometric waveguide dispersion, D_g, using the approximate expression D≈D_m+D_g [18]. The material dispersion (silica in this case) is an input of the problem. The geometric dispersion of the fiber is given in terms of the geometric modal effective refractive index, n_g, as

The evaluation of n_g is carried out by considering that the material refractive indices are wavelength independent. Therefore, its dependence on the wavelength arises from the fiber geometry exclusively. The interplay between the fixed silica dispersion and the tunable geometric dispersion allows us to control the dispersion properties of this new structure. For this reason we have systematically studied the dependence of D_g in terms of the geometric parameters, Λ and a. Following a design procedure similar to that described in Ref. [12], which is based on the above approximate expression for D, one can obtain approximate values for Λ and a providing a desired constant dispersion profile. This part defines the design phase. Of course, we can evaluate D in an exact manner, including the material dispersion, as

and, then, starting from the approximate values for Λ and a, we can fine tune these parameters to obtain the expected dispersion behavior.

 

Fig. 4. Positive (Λ=1.170 µm and a=0.266 µm), nearly-zero (Λ=1.190 µm and a=0.248 µm), and negative (Λ=1.210 µm and a=0.232 µm) flattened dispersion curves near 0.8 µm.

Download Full Size | PDF

 

Fig. 5. Positive (Λ=4.900 µm and a=0.115 µm), nearly-zero (Λ=4.210 µm and a=0.094 µm), and negative (Λ=3.600 µm and a=0.082 µm) ultraflattened dispersion curves near 1.55 µm.

Download Full Size | PDF

Using this procedure, we have studied the dispersion properties of different silica-air fiber designs in two different wavelength windows, the first one located around 0.8 µm and the second one in the vicinity of the optical communication window (around 1.55 µm). We would like to emphasize that all results refer to the fundamental mode of a single-mode structure in the upper forbidden band that corresponds to a polarization doublet HE₁₁. It should be stressed as well that the high accuracy required to calculate D is fully provided by our modal method.

In Fig. 4 we present some examples of positive (anomalous), nearly-zero, and negative (normal) flattened dispersion designs at 0.8 µm. All of them show a zero third-ordered dispersion point. Note that the geometric parameters of the blue curve in Fig. 4 just correspond to that of the plots in Figs. 2 and 3. It is pretty clear the tunability of the structure, even in the region well below the critical silica zero-dispersion wavelength (1.3 µm). It is specially remarkable the possibility of obtaining a flattened positive dispersion profile centered around 0.8 µm (red curve) that can facilitate the stabilization of ultrashort soliton pulses generated at this wavelength (for example with a Ti:Sapphire laser) by a more effective suppression of higher-order dispersion terms.

High-index-core Bragg fibers can also be designed to achieve an ultraflattened dispersion behavior around 1.55 µm. Figure 5 shows the curves for the dispersion coefficient D corresponding to three different Bragg configurations. Note that this ultraflattened behavior is preserved in a large wavelength window that extends over several hundreds of nanometers and, unlike flattened dispersion, permits to obtain a point with zero fourth-order dispersion. The tunability of the structure is also clearly demonstrated by the fact that these designs own positive, negative, and zero D.

 

Fig. 6. Dispersion (solid curves) and relative dispersion slope (broken curves), defined as RDS=(dD/dλ)/D, corresponding to three different selections of the structural parameters to achieve zero four-ordered dispersion at 1.55 µm: red curve (Λ=4.710 µm and a=0.090 µm), blue curve (Λ=4.570 µm and a=0.094 µm), and green curve (Λ=4.465 µm and a=0.096 µm).

Download Full Size | PDF

If we go one step further, we can also recognize diverse intermediate situations in which D has a low value and at the same time the four-ordered dispersion coefficient is null at 1.55 µm. Some solutions for such a challenging task are plotted in Fig. 6. In particular, we would like to point out that the examination of the blue curves in Fig. 6 reveals a fiber design with an ultraflattened dispersion regime and a low dispersion value around 1.55 µm presenting a point of zero third- and fourth-ordered dispersion at this wavelength simultaneously. This is a fascinating result. However, one has to take into account that in our approach light is guided in silica, not in air, and consequently, in principle, it suffers from residual absorption loss and nonlinear effects as four-wave mixing. The performances of our optical proposal finally depend on the particular application.

4. Conclusions

In this paper we have discussed the dispersion properties of high-index-core Bragg fibers. We have numerically demonstrated the ability of these structures to show, for some specific designs, a flattened dispersion behavior (one point of zero third-order dispersion) around 0.8 µm and even an ultraflattened behavior (one point of zero fourth-order dispersion) around 1.55 µm. Moreover, we have recognized some configurations exhibiting positive, negative, or nearly-zero constant dispersion in both wavelength windows. Finally, a noteworthy fiber design that combines low and nearly-constant chromatic dispersion about 1.55 µm with zero third- and fourth-order dispersion at 1.55 µm has been identified.