We propose an engineered microfiber with nano-scale slots that produce ultra-flattened and low dispersion of ±10 ps/(nm·km) over a 340 nm wavelength range. It is comparable with the results in photonic crystal fibers and planar slot waveguides, but can be hardly realized in conventional circular microfibers. By confining the light in a low nonlinearity air slot, the nonlinear coefficient can be greatly reduced. With the unique geometry and excellent performance, the slot microfiber offers large potential in miniature fiber devices for high-speed telecom applications.
© 2011 OSA
The task of controlling the chromatic dispersion is a very important problem in designing practical optical communication systems and dispersion related devices. Low-chromatic-dispersion light-guiding can be essential to maintain linear transfer function for low distortion of analog or digital signals and high-bandwidth data transmission . A key challenge for optical fibers and integrated waveguides is to realize a very flat and low dispersion over a wide wavelength range. Flattened near-zero dispersion can be achieved in photonic crystal fibers (PCFs), with artificially-periodic cladding consisting of micrometer-sized air-holes allowing the flexible tailoring of the dispersion curves. Many novel designs have been reported such as using a structurally-simple PCF with a defected-core , designing a PCF containing elliptical air-holes , selectively filling some of the air holes with oil  and varying the diameters of the air-hole rings along the fiber radius .
For conventional standard single mode fiber (SMF), the main method of dispersion modification is to taper it and change the radius. The dispersion of the tapered fiber depends strongly on the diameter of the taper waist and it can be reduced or even made anomalous at visible and near-infrared wavelengths. The zero-dispersion wavelength (ZDW) can be shifted from 1270 nm to wavelength below 700 nm. At the same time, the dispersion curve is bending and a second ZDW appears at the near-infrared wavelengths [6–8]. However, it is still a difficult task to achieve dispersion flattening even in air-cladding sub-wavelength tapered SMFs (i.e. circular microfibers (CMFs) or optical fiber nanowires) [8–10]. Furthermore, there are strong nonlinear optical properties due to the tight light confinement in the solid microfibers.
The recent advancement of micromachining technology provides new impetus to modify the geometry and performance characteristics of CMF based devices with great flexibility. Typically, micromachining technologies such as focused ion beam (FIB) can be applied to mill CMFs with different geometry, for example, Fabry-Perot cavity with an open notch in a CMF [11,12], ultra-short CMF grating with deep periodic grooves, fiber-top cantilever [13,14], sub-wavelength light confinement tip . Combining current mature technology of drawing CMFs and micro-machining, it is possibly easier to realize a so-called fiberized slot waveguide (FSW), a microfiber with a slot inside. Same as the air-hole in PCF, a slotted structure could provide extra design freedom to tailor the dispersion while keeping a large fraction of the guided mode in a nanometer-wide slot (usually air with low nonlinear coefficient). In this paper, we propose and theoretically investigate the dispersion and nonlinearity of the engineered slot-microfibers. Our simulation results show that low dispersion of ±10 ps/(nm·km) over a 340 nm wavelength range can be realized with highly efficient nonlinearity reduction. It presents great potentials in fiber micro-devices for high-speed optical communications, for example, resonator, grating and coupler. Here, we investigate not only single-slot but also double-slot microfiber because multi-slot waveguides have been extensively studied and show significant performance and advantages. However, our calculation indicates that more slots do not take more benefit on dispersion tailoring and nonlinearity reduction but with more inconvenience in fabrication and handling.
2. Waveguide structure and numerical model
A slot-microfiber can be fabricated by introducing one or multiple slots in a microfiber as illustrated in Fig. 1 (a) and (b) . As a hybrid of a microfiber and a slot waveguide, this kind of device has lots of advantages including larger evanescent field, compact size, flexible design, simplified fabrication and low-loss interconnection to other optical fibers by maintaining the original size at its input/outputs.
The microfiber can be obtained by adiabatically tapering a standard single mode fiber with current mature heat-and-drawing technology [9,10]; the slot can be milled with high accuracy by typical micromachining technologies, such as FIB milling . The method is time-saving and can achieve good sharpness of the slot edge. Of course, it can also be fabricated by re-tapering a thick slot microfiber by femtosecond laser micromachining. It is very flexible to design different slot shapes. Here, we assume n f and n c to be the indices of fiber and coating material (air or low index polymer), W, t, r, and d (d = 2r) to be the width of the slot, the pitch between two slots, the radius and diameter of the microfiber, respectively. As the depth of the slot (d - h) contributes little to the simulation result, we set h to be 0.2d in this paper. A full vector finite element method is used and all simulations shown here are for the quasi-TE polarization in air (n c = 1). The field intensity distribution of quasi-TE mode is shown in Figs. 1 (c) and (d) for the single-slot microfiber and the double-slot microfiber. It shows that the guided light is mostly confined in the slot.
3. Dispersion of slot-microfibers
The total dispersion can be calculated from 
where c is the velocity of light in vacuum, λ is the wavelength and n eff is the effective refractive index. The material dispersion given by Sellmeier’s formula  has been taken into consideration in the calculation.
The dispersion profile of a CMF as a function of wavelength with different diameters is shown in Fig. 2 . The dispersion value changes quickly with the wavelength and it is very difficult to achieve a low dispersion over a wide wavelength only by tuning the radius.
However, the slot microfiber, like the planar slot-waveguide, has additional degrees of freedom in design based on the slot size as compared to the CMF, and it is possible to realize ultra-flattened low dispersion over a wide wavelength range. Figures 3 and 4 shows how the dispersion profile evolves by tuning different geometry parameters in a single-slot and double-slot microfiber, respectively.
For a single-slot microfiber, Figs. 3(a) and (b) shows the dependence of the dispersion profiles on the slot width W and the fiber diameter d, respectively. As W increases from 98 to 108 nm, the dispersion value increases and the two ZDWs shift toward each other and disappear when W > 105 nm. As d increases from 1520 to 1920 nm, the dispersion value increases and the two ZDWs appear when d > 1540 nm and shift away from each other. The flattened nearly-zero dispersion curve appears around W = 100 nm and d = 1.6 μm in Fig. 3(a) and Fig. 3(b), respectively. Figure 3(c) shows two selected flattened near-zero dispersion profiles. One has low dispersion of ±10 ps/(nm·km) over bandwidth in excess of 340 nm from a wavelength of 1320 nm to 1660 nm. Another one has low dispersion value of −5 to 10 ps/(nm·km) over a 340 nm wavelength range, from 1300 nm to 1640 nm.
For a double-slot microfiber, Figs. 4(a), (b) and (c) show the dependence of the dispersion profiles on the slot width, fiber diameter and slot pitch, respectively.
According to these simulation results, we find that the dispersion values drop as the width W increases or diameter d decreases which is similar to the conclusion we draw from the single-slot microfiber. The pitch t has the similar influence on the dispersion profile with d but the first ZDW in Figs. 4(a) and (b) shifts in the opposite direction. Figure 4(d) shows three optimized dispersion profiles by tuning these parameters, with low dispersion of ±10 ps/(nm·km) over bandwidth in excess of 300 nm from a wavelength of 1500 nm to 1800 nm. Compared with the results in the single-slot and double-slot microfibers, there is no big difference except the red-shift of nearly-zero dispersion region in the double-slot microfibers.
4. Highly efficient nonlinearity reduction
Optical nonlinearity can significantly degrade the information capacity and limit the channel power in high speed and long range optical communications. However, CMF displays strong nonlinearity because light can be confined to a very small nonlinear region allowing the ready observation of nonlinear interactions, such as supercontinuum generation, at relatively modest power levels. For an air-slot microfiber, since part of the light is confined within the air region with ultra-low nonlinearity and low refractive index, the nonlinear coefficient (γ) of the waveguide can be reduced.
The nonlinear coefficient in emerging waveguides with subwavelength structures can be found as 
where n and n 2 are the spatial distribution of refraction index and nonlinear-index coefficient. Here we choose nonlinear-index coefficient n 2 = 2.6 × 10−20 m2/W for silica  and n 2 = 5.0 × 10−23 m2/W for air . e and h stand for the spatial distribution of electric and magnetic field, ε 0 and μ 0 are the electric permittivity and magnetic permeability of vacuum, respectively, k is the wave vector and is the unit vector of the propagation direction.
Figure 5 depicts the nonlinear coefficient of a CMF with different diameters as a function of wavelength. It is obvious that with smaller diameter we can obtain larger nonlinear coefficient value because the field intensity of the fiber becomes larger as the diameter decreases.
In Figs. 6(a) and (b) , we plot the nonlinear coefficient profiles at different d and W in a single- and double-slot microfiber. It shows that the nonlinear coefficient value decreases with the increasing of the slot width or fiber diameter because more light is confined in the low nonlinear-index coefficient air-slot region and this part of light makes little contribution to the nonlinear coefficient. On the other hand, it can be seen that the nonlinear coefficient value can drop to ~40% for a single-slot microfiber and ~55% for a double-slot microfiber, compared to a CMF with the same diameter. For the slot microfibers with flattened nearly-zero dispersion as shown in Figs. 3 and 4, the nonlinear coefficient values have the same low level. It means that both the low nonlinear coefficient and flattened nearly-zero dispersion can be realized in a slot microfiber at the same time, which is difficult in a solid-core PCF.
5 Discussion and conclusion
A slot microfiber can be seen as one kind of simple microstructure fiber. It is an engineered microfiber with nano-scale slots that produces both ultra-flattened nearly-zero dispersion and highly efficient nonlinearity reduction. Low dispersion of ±10 ps/(nm·km) over bandwidth in excess of 340 nm can be realized in a slot microfiber which is similar with the results in PCFs and planar slot waveguides, but not possible to be realized in a conventional CMF. By confining the light in an air slot, low nonlinearity can also be achieved keeping the low flattened dispersion, which is different from solid-core PCFs. With the unique geometry and excellent performance, the slot microfiber is helpful for reducing pulse distortion and offers great potential in miniature fiber devices for high-speed telecom applications. Not like the PCFs, the slot microfiber typically is not long, but much longer than the planar slot waveguide. It is perfect for the basic element of different kinds of microfiber devices such as slot-microfiber based resonator, grating, coupler and those slot-microfiber based devices will be useful in many important applications such as femtosecond pulse propagation and ultra-wideband operation. Moreover, we compare the single-slot and double-slot microfibers. The simulation result shows that a single-slot structure is enough for achieving optimized dispersion and nonlinearity performance. More slots do not take more benefit. Considering the flexibility of the fabrication and handling, more slots are not preferred in practical applications.
This work is supported by National 973 program under contract nos. 2010CB327800 and 2011CBA00205, NSFC program no. 11074117 and 60977039, Natural Science Foundation of Jiangsu Province of China under contract no. BK2010247. The authors also acknowledge the support from New Century Excellent Talents program and Changjiang scholars program.
References and links
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