1. Introduction

Confined light guidance through a liquid would provide a basic platform to investigate the fundamental underlying nature of liquid-light interactions, including optofluidic mechanisms, aqueous photochemical processes, and hyper-sensitive spectroscopic sensing of the liquid and its constituents [1–4]. Water and alcohols would be the most basic liquids for these types of applications because they comprise the vast majority of biological or chemical solutions. Despite the high practical potential, design and fabrication of photonic waveguides whose cores are made of these liquids has faced the key technical challenge of finding a proper cladding material whose refractive index is sufficiently lower to support wave-guiding. The conventional waveguide cladding materials such as silica and polydimethylsiloxane (PDMS) [2,3], have significantly higher refractive indices than water and alcohol. In contrast, high refractive index liquids such as CS₂, nitrobenzene, and CCl₄, have been readily integrated into optofluidic waveguides as index-guiding core materials for use in optical amplification and nonlinear optical processes [4,5].

Attempts have been made to guide light through a water core using unconventional cladding materials such as ice [6] and Teflon [7], whose refractive indices are lower than that of water. However, for water-core/ice-cladding waveguides, special thermoelectric cooling was required, and the length of the final waveguide was limited only to a few centimeters. In the case of the Teflon AF cladding waveguide, the refractive index difference between the water core and the Teflon AF cladding was so high that light guidance is limited only to the multimode regime. In practical application of optical nonlinearity, this would cause detrimental dispersion penalties. Moreover, both of these waveguides also suffer from a fundamental incompatibility with conventional silica optical fibers, significantly restricting practical applications.

On the other hand, by permitting injection of liquid at their central hole, defectless silica photonic crystal fibers (PCFs) could provide a feasible alternative for low-refractive-index liquid cores [8,9]. The effective cladding index of these PCFs can be flexibly controlled by adjusting the hole diameter (d) and pitch (Λ) of their air hole arrays in the cladding [10,11]. Moreover, in recent years various techniques for selectively filling the holes in hexagonal PCFs with liquids have been rapidly developed and are now routinely used in the laboratory environment [5,8,12]. When the central hole of a PCF is selectively filled with liquid, the incident light propagates through the liquid directly by either photonic bandgap guidance or effective index guidance, depending on the air-silica lattice structure. In the case of photonic bandgap guidance, the light transmission is limited only to discrete and relatively narrow transmission bands, whose spectral ranges might not include the region of interests for a given type of liquid or application [13]. In addition, the dispersion properties within the transmission band might impose fundamental limitations in nonlinear optical applications.

By combining the advantages of the index-guiding mechanism and the selective filling technique, a defectless hexagonal photonic crystal fiber whose central hole is filled with liquid can provide a viable solution to the shortcomings and limitations of previously mentioned attempts at liquid wave-guiding. With effective index guidance, this method offers a broad and continuous spectral range of operation while guaranteeing flexible control over dispersion properties and relatively low loss. Also, liquid-light interactions occur through the liquid directly, rather than via evanescent wave interaction, which significantly increases the interaction strength by several orders of magnitude. Numerical modeling of this type of liquid core PCF (LCPCF) has been reported for nonlinear optical applications of high refractive index liquids [4,14], which made the detailed management of the effective cladding index unnecessary. Recently, Karasawa [14] reported a design of a water core PCF with a specific structural condition (Λ = d + 0.1μm) to investigate its dispersion properties. However, the paper reported neither detailed light modal intensity distributions of guided modes, which would prove that light is guided through the liquid, nor the optimal ranges of waveguide parameters and refractive indices of liquids for this guiding.

In this study, using the vectorial finite element method (FEM), we perform numerical analyses on an index-guiding LCPCF with the central hole filled by a low-refractive-index liquid and report detailed optimal ranges of the waveguide parameters which ensure that light is confined through the liquid in the fundamental mode. We found that if the LCPCF waveguide parameters are out of the optimal range, the fundamental mode is guided along the silica rims in an annular shape, limited to the evanescent liquid-light interaction. Moreover, we found the optimal waveguide parameters of LCPCFs to ensure that the fundamental mode is guided through the liquid core in a wide range of refractive indices from 1.30 to 1.44. We also report detailed analyses of the three specific cases of liquid cores composed of water (H₂O), ethanol (C₂H₅OH), and butanol (C₄H₁₀O), by including their material dispersion contributions. Especially for the water core LCPCF, we investigated not only chromatic dispersion, but also modal intensity distributions, and optical losses, which lend themselves to bio-chemical applications.

2. Structure of liquid core photonic crystal fibers

As schematically shown in Fig. 1(a) and 1(b), the LCPCF structure analyzed in this study is composed of five air-hole rings in a hexagonal arrangement, characterized by three parameters, the hole diameter d and the pitch Λ, and the liquid in the core. Note that the central silica defect is absent, in contrast to prior conventional PCFs, but a hole with the same diameter as those in the cladding is filled with a low-refractive-index liquid. Here, considering practical issues in the fabrication process such as the longitudinal uniformity of air hole arrays in the PCF, we assume the same hole size and uniform period over the whole cross section.

 

Fig. 1 (a) Schematic diagram of liquid core guidance through the liquid core photonic crystal fiber (LCPCF) (b) Cross-section of the LCPCF with geometrical parameters such as hole diameter d and its pitch Λ. (c) Spectral refractive index of silica (n_silica), liquid (n_liquid) and effective index of fundamental mode (n_HE11) with the condition of liquid core guidance in shaded region

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Once we assume that the liquid-filled central hole serves as the core and the air-silica structure as the cladding of an effective step-index waveguide, we face a very different situation from the prior silica defect PCF. In order to guide the light through the liquid core, the effective index of the fundamental mode (n_HE11) should be less than the liquid core index (n_liquid), which is in turn lower than that of silica, as schematically shown in the grey region of Fig. 1(c). When n_HE11 is higher than n_liquid, the majority of light is guided along the silica region, resulting in a weak evanescent wave interaction between the liquid and the light. Meanwhile, the air-filling ratio d/Λ and hole diameter d should be chosen carefully to provide an appropriate effective cladding index that satisfies n_liquid > n_FSM. For PCFs, the effective index of the fundamental space-filling mode n_FSM of infinite periodic holes in fused silica is regarded as the effective index of the air-silica cladding. Generally, we can lower the n_FSM by increasing the air-filling ratio and decreasing the hole diameter [10,11]. In the following section, by controlling d and d/Λ, we successfully identify the ranges of structural parameters that allow light guidance of the fundamental mode through low-refractive-index liquids, to maximize light-liquid interactions. To the best of the authors’ knowledge, this is the first time such an analysis has been published.

3. Numerical analysis

For numerical analysis of the optical properties of the guided modes, we used the full-vectorial finite element method (FEM) [15]. Using FEM, the effective indices $n_{eff}\left(\lambda \right)=\beta /k_0$were obtained from the eigenvalue equation, Eq. (1) for the magnetic field propagating along the z direction, $\overrightarrow{H}\left(x,y,z,t\right)=\overrightarrow{H}\left(x,y\right)\exp \left[i\left(\omega t-\beta z\right)\right]$.

To analyze the guided mode properties of LCPCFs, the analysis was started with a water core PCF with d/Λ = 0.8 and d = 1.0μm in the spectral range from 0.5 to 1.5μm. In Fig. 2(a), the effective indices n_eff of a few guided modes are plotted as a function of wavelength. Here, the refractive indices of water n_water [16], of silica n_silica [17], and the effective cladding index n_FSM are also shown. In the case of HE₁₁ fundamental mode, the effective index (n_HE11) decreases from 1.39 to 1.27 monotonically and is larger than n_water until the wavelength reaches λ_o = 0.970μm, where we have n_HE11(λ_o) = n_water(λ_o). As the wavelength further increases, the liquid core guidance condition, n_HE11<n_water is fulfilled. In Fig. 2(c), the intensity distribution of the HE₁₁ mode is shown for three cases, (i) λ = 0.5μm (λ<λ_o) where n_HE11(λ)>n_water(λ), (ii) λ = 1.0μm (λ~λ_o) where n_HE11(λ)~n_water(λ), and (iii) λ = 1.5μm (λ>λ_o) where n_HE11(λ)<n_water(λ). At λ = 0.5μm (λ<λ_o), the HE₁₁ mode is mainly guided along the silica region surrounding the water core, but at λ = 1.5μm (λ>λ_o), it is confined to the water core. More detailed evolution of the HE₁₁ mode is shown in Fig. 2(b) as a two-dimensional modal intensity distribution plot in the spectral range from 0.5 to 1.5μm. From Fig. 2(b) and 2(c), we could confirm that the spatial mode redistribution from annular to super-Gaussian occurs gradually, and then becomes a Gaussian distribution as the wavelength increases.

 

Fig. 2 (a) Effective refractive indices of the fundamental and higher order modes of water core PCF with air-filling ratio, d/Λ = 0.8 and hole diameter, d = 1.0μm. (b) Evolution of field distribution of HE₁₁ mode versus wavelength. (c) Comparison of intensity distribution of the HE₁₁ mode at λ = 0.5, 1.0, and 1.5μm. (d) Optical power fraction of HE₁₁ mode in water core, silica, and air-holes versus wavelength.

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This is similar to the behavior of dual-concentric core fibers that are widely used for dispersion compensation [18,19]. Together, all this means that the fused silica region surrounding the water core plays the role of a second concentric core. In this respect, the liquid core PCF could be considered as a three-layer waveguide: the central liquid core, the silica ring that surrounds, and the air-silica cladding, as shown in Fig. 3(a). However, differing from dual-concentric core fibers, the fundamental mode of the LCPCF does not have either a negative dispersion peak at λ_o (Fig. 3(b)) or abrupt optical power redistribution (Fig. 2(b) and (d)). Instead, chromatic dispersion of the HE₁₁ mode in our fiber becomes gradually negative beyond λ = 1.216μm, and the optical power in the water core gradually decreases beyond λ = 1.139μm. Note that both of these wavelengths are longer than λ₀. This could be explained by two arguments: 1) The liquid core diameter of LCPCF is in the order of the wavelength dimension so that its mode field easily and naturally couples to the fused silica region in the longer wavelength range. 2) The large d/Λ value results a narrow silica region surrounding the water core, which reduced impacts on the effective index slope. From this mode field evolution, chromatic dispersion, and power fraction data, it can be concluded that the fused silica region takes the mode coupling effect on the liquid core at wavelengths a little longer than λ_o, and the fundamental mode is eventually guided mainly through the fused silica region at wavelength shorter than λ_o.

 

Fig. 3 (a) Three effective index layers of LCPCFs: the central liquid core (n_liquid), silica ring (n_silica), and the air-silica cladding (n_FSM). (b) Dispersion and dispersion slope of the fundamental mode (HE₁₁) with varied wavelength. (c) Intensity and electric field direction distribution (black arrows) of higher order modes, TE₀₁, TM₀₁, HE₂₁ at λ = 1.0μm, and HE₃₁ at λ = 0.65μm

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The higher order modes shown in Fig. 3(c) are guided through the fused silica region surrounding the water core and interact with the liquid in the central hole only evanescently. As shown in Fig. 2(a), the cut-off wavelength of each mode satisfies the relation n_eff(λ_cut-off) = n_FSM(λ_cut-off) so that the effective index of the guided mode is equal to that of cladding. The corresponding cut-off wavelengths of TE₀₁, TM₀₁, HE₂₁, and HE_31,y, are located at 1.244, 1.215, 1.199, and 0.682μm respectively. We define the x and y degeneracy of each mode by the electric field direction. Also, from Fig. 2 and 3, it is confirmed that the liquid core supports a single-mode, while the concentric fused silica ring core supports multi-modes.

As discussed in Fig. 2, there exists a spectral range which permits guidance of the fundamental liquid core mode, depending on the refractive index of the liquid core and the structural parameters of LCPCF. Therefore, we further investigated the required range of waveguide parameters - the hole diameter d and the air filling ratio d/Λ - of the LCPCF necessary to guide the HE₁₁ mode through the liquid core whose refractive index ranges from 1.30 to 1.44. Numerical results are summarized in Fig. 4.Here, we considered only the waveguide dispersion at the three wavelengths λ = 0.5, 1.0, and 1.5μm. The shaded regions in Fig. 4(a) indicates the range of the air-filling ratio (d/Λ) which allows the fundamental liquid core mode in the LCPCF (n_FSM(λ)<n_HE11(λ)<n_liquid(λ)) when d = 1.5μm, while the shading in Fig. 4(b) indicates the same guidance for the hole diameter (d) when d/Λ = 0.8. For example, if the refractive index of liquid is assumed to be 1.36, the LCPCF allows liquid core fundamental mode guidance at the wavelengths of 0.5, 1.0, and 1.5μm only when d/Λ is larger than 0.940, 0.823, and 0.730, respectively. Likewise, the LCPCF allows the HE₁₁ mode in the liquid core only when d is smaller than 0.623, 1.32, and 2.04μm, for these same wavelengths. It should be noted that as the liquid’s refractive index decreases, more stringent conditions, such as a larger d/Λ and/or smaller d will be required to allow the liquid core HE₁₁ mode guidance in LCPCF. We also express the change of power fraction in liquid core in color map depending on the refractive index of liquid with varied air-filling ratio (Fig. 4(c)) and hole diameter (Fig. 4(d)) at the wavelength of 1.0μm. It is found that the power fraction in liquid core decreases when the structural parameter leaves the region of the fundamental liquid core guidance. Also, the power fraction in liquid core rapidly decreases when the hole diameter is smaller than the wavelength magnitude.

 

Fig. 4 The range of the structural parameters that permits the liquid core fundamental guidance in LCPCF as a function of the liquid refractive index at wavelengths, λ = 0.5, 1.0, 1.5μm: (a) the air-filling ratio (d/Λ) range for the hole diameter of d = 1.5μm and (b) the hole diameter range for the air-filling ratio of d/Λ = 0.8. The shaded regions indicate the fundamental liquid core mode guidance conditions. Power fraction in liquid core at λ = 1.0μm (c) with varied air-filling ratio at d = 1.5μm and (d) with varied hole diameter at d/Λ = 0.8.

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For the next step, we included the specific material dispersion of specific liquids in the waveguide analysis for the spectral range from 0.5 to 1.5 μm, and the results are summarized in Fig. 5. We used the material dispersion data of water (H₂O), ethanol (C₂H₅OH), and butanol (C₄H₁₀O) at room temperature and atmospheric pressure [16,20,21]. Among the three liquids, the refractive index of water is the lowest, while that of butanol is the highest. Figure 5(a) and 5(b) show the range of the air-filling ratio d/Λ for d = 1.5μm and the hole diameter d for d/Λ = 0.8, where the liquid core guidance condition is satisfied. In the same manner, the shaded regions indicate the ranges of the liquid core fundamental mode guidance. For example, if one wants to design an LCPCF with d = 1.5μm which has the liquid core HE₁₁ mode from the wavelength of 0.5μm, the air-filling ratio should be larger than 0.959 for water, 0.938 for ethanol, and 0.894 for butanol as shown in Fig. 5(a). For the LCPCF with d/Λ = 0.8 (Fig. 5(b)), it satisfies the liquid core guidance condition from the wavelength of 0.5μm when d<0.856μm for butanol, d<0.612μm for ethanol, and d<0.508μm for water. Conversely, with d/Λ and d fixed, this means that the lower refractive index of the liquid allows a narrower spectral range for the liquid core fundamental mode guidance. As an example, in the case of d = 1.0μm and d/Λ = 0.8, the HE₁₁ mode propagates through the liquid core for the spectral range λ>0.97μm for the water core PCF, and λ>0.59μm for the butanol core. In Fig. 5(c) and (d), the spectral dependence of the power fraction in water core is shown with varied air-filling ratio and hole diameter respectively. The power fraction in water core gets smaller as the wavelength gets longer because the optical field at a longer wavelength evanescently leaks out to the surrounded air-holes more than that at a smaller wavelength.

 

Fig. 5 The range of the structural parameters that permits the fundamental guidance in LCPCF along the spectral range from 0.5 to 1.5 μm for water, ethanol, and butanol: (a) the air-filling ratio (d/Λ) range for a hole diameter of d = 1.5μm and (b) the hole diameter range for an air-filling ratio of d/Λ = 0.8. The shaded regions indicate the fundamental liquid core mode guidance conditions. Power fraction in water core (c) with varied air-filling ratio at d = 1.5μm and (d) with varied hole diameter at d/Λ = 0.8.

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For the liquid core HE₁₁ modes of water core PCFs, we analyzed in detail the effective refractive index, n_eff(λ) and the chromatic dispersion, D(λ) in the wavelength range λ = 0.5~1.5μm while structural parameters were varied, as shown in Fig. 6. In these analyses, the structural parameters are considered in two dimensions: (i) d/Λ = 0.97~0.77 with d = 1.5μm and (ii) d = 0.45~1.85μm with d/Λ = 0.8. The condition for liquid core HE₁₁ mode guidance, n_FSM(λ)<n_eff(λ)<n_liquid(λ), defined the spectral ranges for the n_eff(λ) curves in Fig. 6. In the case of d = 1.5μm and d/Λ = 0.89 represented as the red curve in Fig. 6(a), the effective index of the liquid core HE₁₁ mode, n_eff(λ = 1.5μm), is 1.2702, which is significantly lower than the refractive index of water n_liquid(λ = 1.5μm) = 1.3168. As the wavelength decreases toward the visible range, n_eff(λ) monotonically increases and eventually crosses n_liquid(λ) of water at λ = 0.859μm. Below this wavelength where n_FSM(λ)<n_liquid(λ)<n_eff(λ), the HE₁₁ mode is no longer confined to the water core but is guided along the silica ring, resulting only the evanescent light interaction as discussed in Fig. 2.

 

Fig. 6 Water core PCFs: (a) Effective refractive index and (c) chromatic dispersion of the HE₁₁ mode for various air-filling ratios with a fixed hole diameter d = 1.5μm. (b) Effective refractive index and (d) chromatic dispersion for various hole diameters with a fixed air-filling ratio d/Λ = 0.8.

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As shown in Fig. 6(d), it could be confirmed that the influence of waveguide dispersion on the chromatic dispersion D(λ) is getting stronger as the hole diameter d is decreased with the fixed air-filling ratio d/Λ, which is the consistent result with the prior reports on PCFs [10]. It is noted that the chromatic dispersion curves consistently blue-shift as we increase d/Λ while keeping the hole diameter constant, so that the zero dispersion wavelength could be tuned over a broad spectral range exceeding several hundred nanometers as in Fig. 6(c). For example, as d/Λ increases from 0.85 to 0.97, the zero-dispersion wavelength blue-shifts from 1.264 to 0.761 μm as shown in Fig. 6(c). Note that the chromatic dispersion value at λ = 1.31μm increases from 19.8 to 146.23ps/nm/km under these conditions.

On the other hand, when we decrease the hole diameter d close to the sub-wavelength range at a fixed d/Λ, the chromatic dispersion starts to blue-shift, and then its slope significantly changes to result in normal dispersion (D<0) over a broad spectral range, as shown in Fig. 6(d). This is mainly due to the waveguide dispersion, and the analyses predicted a large negative dispersion near λ = 1μm region. Also, when the wavelength is near the condition n_eff(λ_o) = n_water(λ_o), dispersion values become more negative regardless of structural parameters. This verifies the fact that a spatial change occurs in the modal distribution, from principally water core to the silica region surrounding it, as in the analysis of Fig. 2. By varying structural parameters, we confirmed the proposed water core PCF can provide efficient dispersion controllability: tuning the zero dispersion wavelengths and changing the sign of the dispersion over a broad spectral range. As an example, for a water core PCF with d = 0.45μm and d/Λ = 0.8, we can achieve negative dispersion along the whole spectral range and a significantly large negative dispersion value of −1040 ps/nm/km at the wavelength of 1.0μm, which could find useful applications in the pulse shaping of laser systems.

We further analyzed the estimated attenuation of water core PCFs. In general, the attenuation of an optical fiber is affected by several factors such as the Rayleigh scattering (RS) loss α_R, the material absorption α_m, the imperfection loss α_IM, and the impurity absorption loss α_im [22]. In this study we considered only α_R and α_m, assuming α_IM and α_im are negligible. The RS loss α_R is proportional to λ⁻⁴ and depends on both the material and the light intensity distribution of the guided modes. For multilayer optical fibers like LCPCFs (Fig. 1(c)), the total α_R is expressed as ${\alpha }_R={\lambda }^{-4}{\displaystyle \sum _i{A}_i{\Gamma }_i}$where Γ_i is the power confinement ratio and A_i the Rayleigh scattering coefficient of each layer [22]. The water core PCF has three layers: the central water core, the silica ring region, and the air-silica cladding, and the corresponding A_i’s are known to be 0.49, 0.86, and 0.16dB/km/μm⁴ respectively [23,24]. We consider Γ₁ for the water core layer as the power confinement ratio of the HE₁₁ mode within 0 ≤ r ≤ d/2, where d is the hole diameter. After calculating the effective core radius r_eff [25], Γ₂ for the silica-ring layer was analyzed within d/2 ≤ r ≤ r_eff, while Γ₃ for the air-silica cladding was analyzed outside r_eff. On the other hand, α_m can be obtained from the extinction coefficient κ by${\alpha }_m=4\pi \kappa /\lambda$. The extinction coefficient indicates the amount of absorption per unit length and comes from the complex refractive index, $\tilde{n}=n+i\kappa$. By applying the complex refractive index of water [26] to the FEM analysis, the effective complex index of refraction ${\tilde{n}}_{eff}$ can be obtained, and from this κ_eff and corresponding α_m are estimated. Note that the contribution of silica to the total absorption is orders of magnitude smaller than the material absorption of H₂O in the spectral range of interest [27]. The calculated optical loss spectra of water core PCFs with structural parameters varied are summarized in Fig. 7.In the case of water core PCFs, the optical loss is mainly dominated by the absorption loss peaks near 0.97 and 1.38μm, which are characteristic of OH vibrational absorption bands of water [26]. The evident loss peak at these wavelengths proves the fact that the HE₁₁ mode is effectively guided through the water core as we mentioned above in the power fraction analysis (Fig. 2(d)).

 

Fig. 7 Optical loss of water core PCFs while applying (a) different air-filling ratio with the fixed hole diameter (d = 1.5μm) and (b) different hole diameter with the fixed air-filling ratio (d/Λ = 0.8)

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Because of its wavelength-dependent nature, the RS loss in the visible wavelengths is two orders of magnitude larger than that in the near infrared wavelengths [28]. For example, when d/Λ = 0.8 and d = 0.45μm, the RS loss at λ = 0.5 and 1.5μm is 8.50 and 0.043dB/km respectively. However, these values are negligible in comparison to the H₂O absorption loss even in the short spectral range, and comprise at most only 17.7% of the absorption loss. It is noted in Fig. 7 that the loss can be reduced by decreasing d/Λ or decreasing d. This is mainly due to the light spreading into the silica region surrounding the water core when we apply small d/Λ or d. The minimum loss at 1.38μm was estimated to be about 3.83dB/cm for d = 0.45μm and d/Λ = 0.8. Therefore, it is found that there exists a critical trade-off between high confinement of light in the liquid core and low optical loss, which should be considered in detailed fiber designs for specific applications. This optical loss analysis based upon the water core PCF can be further generalized to other liquids if the RS coefficients and the complex refractive index information are available.

4. Conclusion

For a uniform hexagonal air-silica photonic crystal fiber (PCF) without a central defect, we analyzed the impact of structural parameters on the fundamental mode guidance through a central liquid core whose refractive index is lower than that of silica. We found that the liquid core PCF is analogous to a three-layer waveguide: the central liquid core, the silica ring that surrounds, and the air-silica cladding, where the coupling between the liquid core and silica ring plays an important role in determining the modal intensity distribution and the chromatic dispersion. This led to a liquid core guidance condition for the HE₁₁ mode such that the modal effective index should be lower than that of liquid. If the condition is not satisfied, then light is mainly guided along the silica region surrounding the liquid core, allowing only the evanescent wave interaction with the liquid. We obtained the optimal ranges of waveguide parameters, d/Λ and d, to guide the fundamental HE₁₁ mode through the liquid core whose refractive index ranges from 1.30 to 1.44. By varying the structural parameters, we confirmed that the water core PCF can provide efficient dispersion controllability: tuning the zero dispersion wavelengths and changing the sign of the dispersion over a broad spectral range. For water core PCFs, we established a technique to estimate the propagation loss by considering the Rayleigh scattering and material absorption, which can be generalized to other liquids.