1. Introduction

Tapered fibers have shown high efficiency to generate white light and supercontinua [1–5], which have many important applications such as pulse compression, spectroscopy, pump-probe measurements, and optical frequency metrology [6–7]. The most important mechanism responsible for spectral broadening is the so-called soliton splitting [8–10], taking place around the zero points of the group velocity dispersion (GVD) and in the anomalous dispersion region where the GVD parameter becomes positive. In principle, the zero dispersion points of the tapered fiber determine the region of spectral broadening, and the GVD characteristics controls the shape and the bandwidth of the broadened spectrum. In order to have a supercontinuum generated in the desired spectral range, one usually needs to design the diameter and the material of the fiber core and the cladding in a way that the zero dispersion point occurs at the appropriate wavelength. Furthermore, we usually desire a homogenous broadening of the spectrum to the largest extent. Therefore, an extremely slow variation with small GVD values in the anomalous dispersion region is of key importance. For photonic crystal fibers, ultra-flat dispersion had been demonstrated by Reeves et al. [11], and the zero-dispersion wavelength of the GVD can be determined by the structure of the fiber. However, for tapered fibers, such a GVD design has not been realized so far. For example, to extend or shift the generation of new spectral components to the infrared, covering for instance the telecommunication window from 1.3 µm to 1.55 µm, a much larger diameter fiber (over 10 µm) would be required for a tapered quartz fiber. Such a thick fiber will result in much lower light intensity in the tapered region and hence a considerably reduced nonlinearity for the conversion process. Certainly, the second point of zero dispersion at longer wavelength can also be used for the generation in the infrared [1][4]. In such a case, a much thinner fiber waist (d=1-1.1 µm) has to be used, which means more difficulties in the fabrication process and a lower damage threshold of the tapered fibers.

In this paper, we demonstrate a considerably simple way to control the GVD characteristics of a tapered fiber. We fill the environment of the tapered fiber with a selected chemical liquid which provides a suitable refractive index [12]. By changing the diameter of the tapered fiber and by using different chemical liquids with varying mixture ratios, we optimize the design of the GVD curve.

2. GVD of a tapered fiber

The GVD of a tapered fiber is defined as the second derivative of the propagation constant β with respect to ω, $D=-\frac{2\pi c}{{\lambda }^2}\frac{d^2\beta }{d{\omega }^2}$. When talking about the GVD in the paper, we always refer to the value D. β can be characterized theoretically by solving the propagation equation in fibers [13]:

$\left[\frac{J_m^{\prime }\left(\kappa a\right)}{\kappa J_m\left(\kappa a\right)}+\frac{K_m^{\prime }\left(\gamma a\right)}{\gamma K_m\left(\gamma a\right)}\right]\left[\frac{J_m^{\prime }\left(\kappa a\right)}{\kappa J_m\left(\kappa a\right)}+\frac{n_2^2}{n_1^2}\frac{K_m^{\prime }\left(\gamma a\right)}{\gamma K_m\left(\gamma a\right)}\right]={\left[\frac{m\beta k_0\left(n_1^2-n_2^2\right)}{a{\kappa }^2{\gamma }^2{n}_1}\right]}^2,$,

where κ ²=$n_1^2$ $k_0^2$-β ² ; γ ²=β ²-$n_2^2$ $k_0^2$ ; a is the core radius; n ₁, n ₂ are the refractive indices of core and cladding, respectively; J_m (x) and K_m (x) are the m^th order Bessel function and modified Bessel function, respectively. m=1 corresponds to the fundamental mode HE₁₁ in the waist region.

The GVD curves as a function of wavelength for tapered fibers made out of fused silica SMF28 fibers in air with different diameters are shown in Fig. 1. We used the Sellmeier equation for fused silica for the values of the refractive index n from Smith [14]. The GVD values are quite large in the anomalous-dispersion regime, ranging in the order of 200 ps/km/nm; meanwhile, the first zero dispersion points do not exceed a wavelength of 800 nm for a fiber diameter below 3 µm. This phenomenon, caused by the large difference between the refractive indices of the fiber material and air, limits the position and the extent of spectral broadening. To achieve supercontinuum generation in the spectral range of longer wavelengths and to extend the bandwidth, we should find media with intermediate refractive index values between n_air and n_fiber to fill the environment of the fiber taper. Organic chemical liquids are promising candidates for this task.

 

Fig. 1. The calculated GVD curve of a tapered SMF28 fused silica fiber in air with diameter (I) 1 µm, (II) 1.5 µm, (III) 2 µm, (IV) 2.5 µm, and (V) 3 µm.

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3. Refractive indices and dispersion of some chemical liquids

We measured the refractive indices of acetonitrile, pentane, and hexane in the visible spectral region using an Abbe Refractometer, as summarized in Table 1. Limited by the instrument, we were not able to perform the measurement in the infrared. Equations (1)–(3) are the fitted Sellmeier equations for the three chemicals using the measurement data. For acetonitrile, we combined the measurement data in the infrared (from 1200 nm to 2000 nm) given in Ref. [15] with our measurement in the visible. Unfortunately, no data in the IR are available so far for the other two chemicals.

Table 1. Measured refractive indices and Sellmeier equations of acetonitrile, pentane, and hexane

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Although the Eqs. (2) and (3) for pentane and hexane are sufficiently precise only in the visible range, the dispersion of organic compounds varies only slowly in the infrared, so that we extrapolate them into the longer wavelength region with reasonable accuracy. This is certainly a first approximation to the problem. However, in order to be able to estimate the required fiber waist thicknesses and liquids, this method gives valid results. Comparing the exact refractive index of acetonitrile in the IR (the only component with a tabulated refractive index in the IR) with the extrapolation, we found only small deviations of the GVD (around 10 ps/nm/km). Also, the GVD zero point uncertainty is only in the range of 0.1 µm. Therefore, before more exact measurements of the IR indices of other liquids can be made, this method has to be sufficient. In the future, immersed tapered fibers can be characterized with respect to their GVD using white light interferometry.

In principle, water can also provide an appropriate refractive index (1.33 at 589 nm), and its Sellmeier equation with respect to wavelengths from 0.2 µm to 1.1 µm can be found in Ref. [16]. However, strong absorption takes place in the infrared due to the O-H bond vibration overtones, which excludes water for our application. The transmission curves of a 9.8 mm cuvette with water, acetonitrile, pentane, and hexane in the infrared region are shown in Fig. 2 (corrected for the cuvette reflection). Acetonitrile, pentane, and hexane do not suffer from near infrared absorption since they do not possess O-H vibrational overtone absorption lines. The absorption bands around 1400 nm are on the order of 1 cm^-1 and do therefore influence the refractive index in that region only very slightly.

 

Fig. 2. The transmission curves of a 9.8 mm cuvette of water (dashed), acetonitrile (dotted), pentane and hexane (solid). The curves of pentane and hexane are almost on top of each other.

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4. GVD of tapered fibers immersed in chemical liquids

Figure 3 shows the calculated GVD of tapered fibers with a diameter of 3 µm, assuming that the fibers are immersed in (I) acetonitrile, (II) pentane, and (III) hexane, respectively.

 

Fig. 3. The GVD curve of tapered fiber with diameter 3 µm when it is immersed in (I) acetonitrile, (II) pentane, and (III) hexane.

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The curves in Fig. 3 reveal that a tapered fiber with a diameter of 3 µm possesses favorable GVD properties in the anomalous dispersion region if immersed into these three chemicals. The first zero dispersion point appears around 1 µm in the near infrared. The GVD curve has quite small values (<40 ps/km/nm) over the span between the two zero-dispersion points. The bandwidth of anomalous dispersion of the tapered fiber in acetonitrile (950 nm~1650 nm) is broader than that of the tapered fiber in pentane and hexane, while smaller GVD values (less than 20 ps/nm/km) can be achieved using pentane and hexane. We can further conclude that using acetonitrile, pentane, or hexane as the environment of a tapered fiber, a supercontinuum generation should be feasible in the infrared and the corresponding spectral width should be significantly enlarged. Depending on the exact length of the waist, the continuum might stretch even further into the infrared, as the absorption values are around 1–10 cm^-1.

Harbold et al. reported continuum spectrum generation around the second dispersion zero point (1.26 µm) of a tapered fiber in air with a diameter of 1 µm [4]. In our case, supercontinua should be feasible at even longer wavelengths in the infrared, e.g., around 1.6 µm when using pentane and a fiber diameter around 3 µm.

Regarding tapered fibers with diameters of 2.5 µm and 3.5 µm, GVD curves are shown in Figs. 4(a) and (b), respectively. With smaller diameter, the tapered fiber provides lower and steeper group velocity dispersion, and the fiber immersed in hexane does not reach the anomalous dispersion region. With a larger diameter of 3.5 µm, the anomalous dispersion region covers more wavelengths. The first zero-dispersion wavelength (ZDW) appears in a lower position while all the second ZDWs of the three candidates occur beyond 1.7 µm.

 

Fig. 4. The GVD curves of tapered fibers with diameters of (a) 2.5 µm and (b) 3.5 µm when they are immersed in (I) acetonitrile, (II) pentane, and (III) hexane.

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5. Optimization of the GVD curve

5.1 Mixing of different chemicals

In some cases, the center frequency of the pump cannot exactly match the ZDW of the GVD curve, which results in lower intensity and narrower spectra of the supercontinuum light. Therefore, we mix different chemicals to get a new effective refractive index. This allows the optimal ZDW to be generated just at the frequency of the pump pulse by changing the ratio of the chemical constituents. For a pump pulse at 1.55 µm, the second ZDW can occur exactly around 1.55 µm when a tapered fiber with a diameter of 3 µm is surrounded by pentane and hexane mixed in a ratio of 1:1, as shown in Fig. 5. The curve in Fig. 6 shows the variation of the second ZDW position with the change of the ratio between pentane and hexane. A tuning range from 1.38 µm to 1.7 µm for the second ZDW can easily be achieved by varying the mixing ratio.

 

Fig. 5. The GVD curve of a fused silica tapered fiber with a diameter 3 µm when it is immersed in a mixture of pentane and hexane (1:1).

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Fig. 6. The second zero-dispersion wavelength position versus the ratio of hexane in the mixture (hexane and pentane) when the fused silica fiber taper diameter is 3 µm.

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5.2 Selection of the fiber material

The diameter of tapered fibers is one of the significant factors which influence the nonlinear effects in the fiber. However, because of the limited range of the refractive indices that the chemicals can offer, the diameter of the tapered fiber cannot be smaller than 3 µm in order to achieve an anomalous dispersion range. For the first ZDW of GVD curve, no matter what liquid is employed and what diameter of the fiber taper is used, the first ZDW cannot be generated in the telecommunication window. Therefore, we would like to select some alternative fiber materials (BK7, SF6, and SF9) instead of fused silica to serve as optical fiber material [17] which could overcome such disadvantages.

For the calculations of the GVD, the necessary formulas for the three selected glasses, such as the Sellmeier equations, are given in Ref. [18], and their basic optical properties are shown in Table 2 [19]. The GVD curves of a tapered fiber made of BK7 show the desired anomalous dispersion properties when the fiber taper with a diameter of 2 µm is immersed in acetonitrile, pentane, and hexane, respectively (see Fig. 7). Figure 8 shows the GVD curves generated by tapered fibers made out of SF6 and SF59, immersed in chlorobenzene (refractive index 1.525 at 589 nm). Their first ZDW can occur in the telecommunication window and their GVD characteristics are rather flat. The advantage of such high-index glasses is to allow a much greater range of immersion liquids with refractive indices that are smaller than the glass index.

Table 2. The optical properties of glasses Fused silica, BK7, SF6, and SF59 (n₂(esu)=174 n₂(cm²/W))

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Fig. 7. The GVD curve of a BK7 tapered fiber with diameter 2 µm when it is immersed in (I) acetonitrile, (II) pentane, and (III) hexane.

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Fig. 8. The GVD curve of a tapered fiber immersed in chlorobenzene, with a fiber taper material of (I) SF6, d=3 µm, (II) SF59, d=3 µm, (III) SF6, d=4 µm, and (IV) SF59, d=4 µm.

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6. Conclusions

Theoretical calculations reveal that the design of the GVD in tapered fibers can be tailored when we immerse the fibers in transparent liquids (such as acetonitrile, pentane, and hexane). The GVD curves exhibit slow variations and small values (<40 ps/km/nm) in the anomalous dispersion region. The second zero-dispersion wavelength of the GVD curve can be generated in the infrared (e.g., around 1.55 µm). When mixing pentane and hexane, the ZDW can be fine-tuned to match the center frequency of the pump pulse around 1.55 µm, which can optimize supercontinuum generation. Finally, by use of alternative fiber materials (BK7, SF6, and SF59) for the tapered fiber and immersing them into chlorobenzene, we can reduce the diameter of the fiber taper to 3 µm and position the first ZDW of the GVD curve in the telecommunication window.