## Abstract

We propose a method to control the chromatic dispersion properties of photonic crystal fibers using the selective hole filling technique. The method is based on a single hole-size fiber geometry, and uses an appropriate index-matching liquid to modify the effective size of the filled holes. The dependence of dispersion properties of the fiber on the design parameters such as the refractive index of the liquid, lattice constant and hole diameter are studied numerically. It is shown that very small dispersion values between 0±0.5ps/nm-km can be achieved over a bandwidth of 430–510nm in the communication wavelength region of 1300–1900nm. Three such designs are proposed with air hole diameters in the range 1.5–2.0*μ*m.

©2006 Optical Society of America

## 1. Introduction

Index-guiding photonic crystal fibers (PCF), also called microstructured fibers, have been extensively investigated because of their unique properties compared to bulk media and conventional fibers. The great variety of possible geometries in microstructured fibers offers the possibility to control their transmission, dispersion and nonlinear properties, leading to a plethora of applications. In particular, the dispersion properties of index-guided PCFs can be designed to match various purposes. Initially, the basic design of a microstructured fiber, characterized by its core size, pitch and a (uniform) hole diameter, was studied in detail experimentally and theoretically (see e.g. [1, 2, 3]), especially with the aim to achieve ultra-flat dispersion [4] over wide range of wavelengths.

To achieve even lower values of dispersion and to improve the dispersion flatness over wider bandwidths, more complicated designs with various core geometries [5] and/or multiple hole diameters [6, 7], and even hole shapes [8] have been studied. Wider design freedom makes it possible to target several fiber properties, such as the chromatic dispersion and the mode area [9, 10, 11], in the same design. The potentially very large parameter space of possible designs invites the use of sophisticated optimization approaches [12]. Saitoh et al. [7] have proposed a dispersion-controlling technique that uses four to five different sized air holes with diameters as small as 0.5*μ*m in the cladding. They showed that it is possible to design a PCF with flat dispersion of 0±0.4ps/nm-km over the wavelength range 1230–1720nm. Similarly, Wu et al. [13] have numerically demonstrated a dispersion-flattened PCF with dispersion of 0±0.25ps/nm-km over the range 1295–1725nm by using two different sizes of air holes with the smallest air hole diameter of about 0.5*μ*m.

Besides changing the geometry of the microstructured or photonic-band-gap fiber, an alternative method to control its transmission and polarization properties is by filling the air holes, either completely or selectively, by various liquids such as water [14], ethanol [15], polymers [16, 17, 18], and liquid crystals [19, 20].

In this paper, we propose a selective hole-filling technique for controlling the dispersion of a PCF to achieve ultra-flattened dispersion over wide wavelength regions. This approach provides a couple of advantages. First, our staring point is a relatively simple holey fiber with a uniform geometry, and a rather large hole diameter, that is easier to fabricate compared to fibers with multiple different sub-micron hole sizes. Second, the selective hole filling makes it possible to adapt the same basic microstructured fiber design to various applications thus providing us with greater flexibility. We note that the same dispersion controlling technique can be used to design a PCF for dispersion compensation, birefringence control, and supercontinuum generation optimization (by controlling the zeros of group velocity dispersion [21]) among others. The refractive index of the liquids considered in this paper is lower than the index of the glass and the guidance is due to total internal reflection. Alternatively one could fill the air holes with high index fluids. These belong to the so-called class of ARROW PCFs [22, 23] where the guidance is due to anti-resonant reflection and provide another means to control the dispersion.

There are several techniques for selectively filling the holes with liquid. If the PCF consists of different air hole sizes, the flow speed of the liquid, which is larger for large holes, can be exploited [24, 25] to selectively fill the holes. One could use the fusion splicing technique with tailored electric arc energies and fusion times to selectively fuse [14, 15, 26] the outer rings of the PCF. This method can be empolyed to fabricate the fiber proposed in this paper. A greater control can be achieved by selectively blocking the unwanted air holes with photolithographic masking technique [27] or with epoxy [17]. Undoubtedly, these techniques will be greatly improved in the future, thus opening new possibilities in specialty fiber design. The above references demonstrate filling lengths of over tens of centimeters, which are sufficient enough for applications like optical tunable filters, variable optical attenuator, tunable optical waveguides [16] and supercontinuum generation applications [21].

## 2. Liquid-filled photonic crystal fibers and analysis method

The idea central to this paper is to work with microstructured fiber whose air holes are all of the same size, and to mimic the effects of varying certain hole-sizes by filling these holes with a liquid of an appropriate index of refraction and chromatic dispersion. The concept of the scheme is depicted in Fig. 1. Filling an air hole with liquid effectively reduces its diameter, depending on the refractive index of the liquid. The fabrication of such a fiber is simplified due to large air hole diameters and the uniformity of the air holes. The inner rings of the air holes can be filled with liquid, first by fusing the outer rings of air holes with fusion splicing technique [26] and then by immersing one end of the fiber in a liquid reservoir and applying vacuum to the other end of the fiber[16].

Although a variety of filling configurations can be studied, including asymmetric filling to produce birefringence effects, in this paper we consider only filling the entire rings of air holes. The design studied in this paper consists of a PCF with four rings of air holes with the central air hole missing. The inner two rings of air holes are filled with a liquid (see Fig. 1). Using different liquids, and optimizing the geometry of the holey fiber matrix, various chromatic dispersion “landscapes” can be readily achieved.

We analyze this structure using a full vectorial finite element method (FV-FEM) [28]. A triangular mesh [29] is used to discretize the geometry and the transverse and longitudinal electrica field components are represented in terms of the Whitney elements and the first-order node elements respectively. As the confinement losses are low in our design, the boundary condition becomes inconsequential for the purpose of computing dispersion. The total dispersion (*D*) is computed in two steps. First, the effective index of the fundamental mode, *n*
_{eff}(*λ*), is calculated over a dense grid of wavelengths using FV-FEM. Then, the dispersion is evaluated as

where *c* is the velocity of light in vacuum and the second derivative is computed using a 3-point difference rule.

## 3. Optimization procedure and results

In order to manage the multi-dimensional parameter space consisting of the refractive index of liquid *n _{L}*(

*λ*), the pitch Λ, and the air hole diameter

*d*, the optimization is performed heuristically in two stages. In the first stage we consider a hypothetical liquid that has a constant, wavelength independent refractive index. However, the material dispersion of the fiber is taken into account, and the refractive index of silica glass is calculated using the four-term Sellmeier formula throughout the optimization. The reduced set of parameters, namely the refractive index of the hypothetical liquid

*n*, Λ and

_{L}*d*are then optimized to achieve ultra-flat, near zero dispersion. In the second stage we choose an appropriate liquid such that it has a refractive index close to the optimized value

*n*in the wavelength region of interest. As we will see, the contribution of the material dispersion of the liquid to the total dispersion is significant. Hence, we desire a liquid that has a low dispersion. Once a liquid is chosen we re-optimize the design by adjusting both, Λ and

_{L}*d*. The advantage of working initially with an artificial liquid is that one doesn’t need to choose a liquid apriori from the great number of available index-matching liquids. We note that this is not an exhaustive search over the whole parameter space. A more refined optimization, such as the one used in Ref. [12], would likely provide even better solution. Nevertheless, our approach serves as a simple practical method to design a dispersion flattened fiber.

Next, we illustrate the first stage of the design optimization. Instead of retracing all the steps taken during the optimization, we first illustrate the effect of varying one of the design parameters on the dispersion curve while the rest are kept constant. This is done in the vicinity of the optimized parameters as it gives the added information about the sensitivity w.r.t. variations of the parameter values. We then describe the optimization procedure employed to calculate the final optimal design parameters.

Figure. 2 shows the effect of changing *n _{L}*, on the total dispersion. One notices that both, the magnitude and slope of the dispersion are modified as

*n*, is varied. From Fig. 3 we see that as the value of the pitch is increased, the total dispersion increases without modifying its slope drastically. The effect of varying the air hole diameter

_{L}*d*is depicted in Fig. 4. One observes from this figure that increasing

*d*has the effect of increasing the negative slope of the dispersion without significantly modifying its value. Thus, varying Λ and

*d*have the desired effect of modifying mostly only one of the dispersion value or the dispersion slope, while varying

*n*modifies both. Hence, we took the following optimization procedure. Starting from arbitrary parameters (we chose

_{L}*n*, = 1.3, Λ = 1.50

_{L}*μm*and

*d*= 1.35

*μm*) the value of

*n*, is varied until we have a flat dispersion although not necessarily near zero. Then we successively change Λ and

_{L}*d*to either raise/lower the dispersion or to modify its slope. Following this procedure one arrives at Fig. 5 which shows ultra-flat zero dispersion in the wavelength region of 1400nm–1800nm The optimal refractive index of the hypothetical liquid is found to be 1.3975. Figures 2, 3 and 4 give us an indication of the sensitivity of design parameters. For example for the dispersion to be in between 0 ± 1.0ps/nm-km over 400nm wavlength range the allowed tolerance in the air hole diameter and the lattice constant are, respectively, ±5% and ±1% of the optimal values, which are comparable to the previously reported values [12]. The thermo-optic coefficient dn/dT of the liquids considered in this paper is of the order of 4 × 10

^{-4}/

*°C*From figure 2 we see that this value is small enough to allow a stable operation over temperature fluctuations of ±3

*°C*. On the other hand it is large enough to allow tuning [30] of dispersion by temperature.

Having obtained a preliminary design using an artificial, dispersionless liquid, we continue with the second optimization stage. We select an oil, we choose to call it oil#1, whose refractive index is given by the Cauchy equation

where the unit of wavelength *λ* is Å. This is the typical characteristics of “Refractive Index Liquid Series AA” oils manufactured by the Cargille Laboratories. The reason this particular oil is suitable for our purpose is that it has a refractive index close to *n _{L}* = 1.3975 in the wavelength window of interest.

Before finalizing the design, it is worthwhile to look at the contribution of the material dispersion of liquid to the total dispersion. Shown in Fig. 6 is a comparison of the fiber dispersion change introduced by the oil#1 with the material dispersion of the oil itself. Since the fundamental mode is strongly confined to the core region we expect that only a fraction of the material dispersion of the oil contributes to the total dispersion. As evident from Fig. 6 this is about 10%. Hence we require that the material dispersion of the liquid be small in the wavelength range of interest in order that our optimized design without the liquid is close to the final design. Fortunately there are many index matching liquids with relatively low dispersion in the communication wavelength region. When nonlinear effects are considered it is reasonable to assume that the contribution of the liquid is small due to the confinement of the mode. Nevertheless the liquid increases the effective area of the mode thereby reducing the effective nonlinear coefficient.

Finally, with the hole-filling liquid chosen, we can return to the re-optimization of the fiber’s geometry. Further adjusting the pitch and the hole diameter results in the dispersion curve shown in Fig. 7. We can see that the selective hole filling can be used to obtain ultra-small and flat dispersion over a very wide region of wavelengths.

To demonstrate the flexibility of the proposed technique, we also show in the figure other optimized designs using two other oils. These oils, called oil#2 and oil#3, are manufactured by the Cargille Laboratories and are classified as “Refractive Index Liquid Series AAA”. The index of refraction of these oils is parameterized as,

where the unit of wavelength is again Å. Although it is not depicted in Figure 7, for each design the magnitude of the dispersion near zero can be adjusted and is related to the bandwidth over which we have small dispersion. The optimized parameters and the dispersion characteristics of the three designs are summarized in Table 1. Compared to design 1, design 3 has its dispersion curve shifted towards shorter wavelengths, while preserving low dispersion values of 0 ± 0.5ps/nm-km over bandwidth in excess of 500nm. The dispersion curve of design 2, on the other hand, is shifted towards longer wavelengths compared to design 1. The comparison of different optimization results also clearly demonstrates that the method can be used to fine-tune positions of the zero-dispersion wavelength, a property important for supercontinuum generation.

## 4. Conclusion

We proposed a selective liquid-filling approach to precisely tune and tailor the dispersion properties of the photonic crystal fibers. This approach results in fibers with large air holes of the same size making their fabrication much easier compared to other designs. The numerical computation shows that the dispersion value and the dispersion slope of these fibers can be controlled by selectively filling a liquid in the air holes and by varying the design parameters such as the lattice constant and hole diameter. To demonstrate the effectiveness of the approach, we proposed three designs of liquid-filled photonic crystal fibers exhibiting ultra-flat dispersion values between 0±0.5ps/nm-km over a bandwidth of 430–510nm in the wavelength range between ~1300 and ~1900nm. Further optimization can extend the wavelength range of zero-dispersion by simply selecting different liquids and PCF structures. This selective liquid-filling approach can also be extremely effective in designing a PCF for dispersion compensation, supercontinuum-generation optimization, birefringence control and other photonic device applications.

## Acknowledgments

This work is supported partly by AFOSR under the grant number FA9550-04-1-0213 and by the SungKyunKwan university through the second phase of the BK21 program.

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