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Abstract
Microstructured optical fibers (MOFs) have attracted intensive research interest in fiber-based optofluidics owing to their ability to have high-efficient light-microfluid interactions over a long distance. However, there lacks an exquisite design guidance for the utilization of MOFs in subwavelength-scale optofluidics. Here we propose a tapered hollow-core MOF structure with both light and fluid confined inside the central hole and investigate its optofluidic guiding properties by varying the diameter using the full vector finite element method. The basic optical modal properties, the effective sensitivity, and the nonlinearity characteristics are studied. Our miniature optofluidic waveguide achieves a maximum fraction of power inside the core at 99.7%, an ultra-small effective mode area of 0.38 µm2, an ultra-low confinement loss, and a controllable group velocity dispersion. It can serve as a promising platform in the subwavelength-scale optical devices for optical sensing and nonlinear optics.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optofluidic techniques have attracted intensive attention for the advantages of trace sample consumption, high sensitivity, fast response, and intriguing light-fluid interactions [1–3]. As a result, numerous optofluidic applications have been proposed, such as biological analysis, particle trapping, and chemical synthesis [4–9]. Most of these applications are implemented in a lab-on-chip configuration via microfluidic channels with dimensions varying from tens to hundreds of microns, which may limit the high-sensitivity detection due to the short length of light-fluid interaction on chip [4–9]. Alternatively, a lab-in-fiber configuration, or fiber-based optofluidics, has been developed to enhance the interaction length and improve the coupling robustness [2,10,11]. In optofluidic applications with a miniature waveguide, a significant enhancement of light intensity is critical to implement the trace fluidic analysis or realize an efficient nonlinear effect [1,6,12–14]. However, it is difficult to build up subwavelength optofluidic waveguides [12]. Previous attempts include the design of subwavelength-scale optofluidic slot waveguides [6,15] and tapered capillary fibers [16,17]. Slot waveguides are capable of confining light in subwavelength-scale channels, however, the short light-fluid interaction length and high coupling loss may hinder their wide applications [4–7]. The tapered subwavelength-scale liquid-core capillary fibers possess a simple structure, whereas they cannot confine light tightly in the fluid with a low refractive index [16,17].
The microstructured optical fiber (MOF) is an important building block in the architecture of optofluidics [18–21]. In the early stage of optofluidic MOF waveguides [19,20], hollow-core MOFs were used as a lab-in-fiber platform by infiltrating the liquid samples in the central hollow core. This is quite unique since the air-silica structure with high air-filling fraction can form a cladding with an average refractive index low enough to support a high-index-contrast optofluidic waveguide for various aqueous solutions in the central core. In the optofluidic MOF, light and fluid can be confined within the same uniform microchannel simultaneously with a spot size around several or tens of microns along the entire fiber length, achieving an ultra-high sensitivity by exploiting the direct interaction of almost the entire light field with the fluid and a long interaction distance [19,20,22–25]. The liquid-core MOF with the minimum microhole size of 2.5 µm has also been used for supercontinuum generation [21]. Due to the large index contrast of the waveguide in the optofluidic hollow-core MOF, it may have the potential to be used in the subwavelength-scale fiber optofluidics with the lowest sample consumption. Despite huge potential in a great deal of promising applications, no comprehensive theoretical studies of the miniature subwavelength-scale MOF optofluidic waveguides hitherto have been conducted, which hinders the development and deployment of this technique.
Here, we propose a tapered subwavelength-scale optofluidic waveguide using a specific hollow-core MOF. The core of the MOF is tapered down to the subwavelength scale with aqueous solution selectively infiltrated. The waveguide features are investigated in detail, including the modal guidance condition and the adiabatic transition condition. Besides, the power fraction in the core and the relative sensitivity coefficient provide a deep insight into the highly sensitive sensing, while the effective mode area, the power density, the nonlinear coefficient, and the group velocity dispersion (GVD) indicate the enhanced nonlinearity of the sub-wavelength optofluidic MOF waveguide. Moreover, the sample consumption, the fluidic sample infiltration duration and fabrication issues, the interface coupling and splicing issues, and the attenuation induced by the absorption of water are discussed to provide a recipe for utilizations. The detailed analyses on the properties and utilizations of the tapered MOF suggest a promising low-loss subwavelength-scale optofluidic fiber platform for various applications such as optofluidic devices, optical sensing, nonlinear optics, etc.
2. Structure of the tapered MOF
In this study, we propose a tapered hollow-core MOF structure with uniform holes in the cross-section and adiabatic tapered microchannels along the fiber, preparing a host channel in the center for a subwavelength-scale optofluidic core as an efficient optofluidic waveguide. As shown in Fig. 1, the proposed MOF is composed of an aqueous core and air-holes arranged in a hexagonal lattice in a fused silica background, characterized by the hole diameter d, the center-to-center pitch Λ, and the number of rings. The waist of the tapered MOF has a subwavelength-scale core, and connects to two original untapered sections via smooth transitions. Instead of using the traditional hollow-core MOF with a large core, the uniform hole design will benefit the tapering design, i.e., when the aqueous core is small, the air-silica cladding can still support a high index-contrast waveguide.
Fig. 1. Schematics of the tapered subwavelength-scale MOF structure, which can be mechanically spliced with a conventional optical fiber in a high quality. Insets are the cross sections of the MOF.
It is known that the optical coupling issue is always critical for various dissimilar waveguides [26–30]. Here the proposed tapered MOF can address the challenge in subwavelength-scale optofluidic coupling. As shown in Fig. 1, the MOF can be mechanically aligned to the conventional fiber with a low insertion loss and good robustness. Moreover, a transition with an adequate length can facilitate an adiabatic transmission for the fundamental mode, and enable the mechanical support for the microfiber handling. This hollow-core MOF tapering architecture provides a powerful platform for the investigation of the optofluidic waveguide in the MOF [28,31].
3. Modelling of the optofluidic MOF
The basic properties of the optical modal guidance in the proposed optofluidic waveguide are investigated by the full vector finite element method (FEM) in this section. We assume that the core of the MOF is tapered down to a subwavelength-scale level, and then the refractive index of the thin air-silica cladding is close to that of air. After the central hole is selectively infiltrated with an aqueous solution, the index contrast between the aqueous core and the cladding is high enough to support an efficient index-guiding in a wide spectral range [12,16,20]. Our study starts with a MOF with four rings, and the air-hole diameter d ranges from 300 nm to 5000 nm. We focus on properties of the uniform waist part. The pitch Λ satisfies the equation Λ = d + 50 nm, i.e., the wall thickness can be fixed at 50 nm, and this thin silica membrane in MOF can be fabricated in experiments [32]. In this way, the air fraction in the cladding is increased to reduce the effective index of the cladding, thus providing high index-contrast waveguide for the aqueous core.
3.1 Modal guidance condition
The modal guidance properties of the aqueous-core MOF are analysed firstly. To be an index-guiding waveguide, the effective refractive index of the guided mode neff should be larger than that of the air-silica cladding (the fundamental space-filling mode (FSM) nFSM) and smaller than that of the aqueous core. Since the refractive indices of most aqueous solutions are similar to that of water [13], we use water as the core material in our study. The material dispersions of water and silica at 20°C and atmospheric pressure were extracted by the Sellmeier equations [33,34], while the refractive index of air was assumed to be 1 for all wavelengths.
As shown in Figs. 2(a, b), the refractive indices of water nwater were adopted as 1.332 and 1.316 respectively at 633 nm (the operation wavelength of He-Ne laser) and 1550 nm (a common wavelength for optical fiber telecommunication), and the refractive indices of silica nsilica were 1.457 and 1.444, respectively. The refractive index of the FSM nFSM can be obtained by applying the FEM on the elementary piece of the cladding (Fig. 2(c)) that acted as a boundless propagation medium [35]. The distribution of FSM varies with the air-hole diameter, and Fig. 2(c) shows the evolution of the FSM when the air-hole diameters are 600 nm, 1000 nm, 1500 nm, 2000 nm, 3000 nm, and 4000 nm, respectively, at 633 nm (Figs. 2(c1-c6)) and 1550 nm (Figs. 2(c7-c12)). At 633 nm, when the air-hole diameters are relatively small, FSMs are confined in the membrane between two air-holes, and only a small part of power leaks into the air holes. As the air-hole diameter increases, the membrane is relatively longer, a larger portion of the FSM extends into air holes, which decreases nFSM. However, when the air-hole diameter is larger than 1500 nm, the mode escapes from the membrane to evolve into the triangular solid area surrounded by three adjacent air-holes, resulting in the increase of nFSM. As a result, the calculated nFSM in Fig. 2(a) decreases first before the core diameter reaching 1500 nm, and then increases. In contrast, as shown in Figs. 2(c7-c12), the light with longer-wavelength cannot be strongly confined in the thin membrane, and a significant portion of the FSM locates in the air-holes. Moreover, the FSM evolves more slowly in the air-holes, and escapes from the membrane at a larger air-hole diameter of around 4000 nm. The calculated nFSM in Fig. 2(b) changes more slowly, and has no obvious increase in the wavelength range in our investigation.
Fig. 2. (a, b) Effective refractive indices of the guided modes in MOFs as functions of the core diameter at 633 nm and 1550 nm, respectively. (c) Evolution of the FSM distribution when the air-hole diameters are 600 nm, 1000 nm, 1500 nm, 2000nm, 3000 nm, and 4000 nm at 633 nm and 1550 nm. (d) Electric field distributions of HE11, TE01, HE21, and TM01 modes in the MOF with a core diameter of 900 nm at 633 nm.
Figures 2(a, b) also show the mode conditions of the MOF with respect to the core diameter. The effective refractive indices of the guided modes in the MOFs are plotted as functions of the core diameter at 633 nm (Fig. 2(a)) and 1550 nm (Fig. 2(b)), which are over nFSM. Only the fundamental mode (HE11 mode) exists when the core diameter was down to a certain value, which corresponds to the critical core diameter (dsm) for the single mode operation. In the case of the wavelength of 633 nm, the fundamental mode guidance starts at the core diameter of smaller than 300 nm, and dsm is around 500 nm (Fig. 2(a)). The calculated nHE11 increases from 1.168 to 1.329 when the core diameter ranges from 300 to 5000 nm, and finally approaches nwater, indicating a tight confinement for the fundamental mode in the aqueous core. The effective refractive indices of higher-order modes, TM01, HE21, and TE01 modes, separate from each other clearly when the core diameter is no more than 2 µm. Figure 2(d) presents the electric field distributions of HE11, TE01, HE21, and TM01 modes in the MOF with a core diameter of 900 nm at the wavelength of 633 nm, showing the confinement of modes in the aqueous core. For the wavelength of 1550 nm, the fundamental mode starts at the core diameter of 550 nm and the higher-order modes appear when the core diameter dsm is 1300 nm. The calculated neff increases from 1.072 to 1.300 as the core diameter increases from 300 to 5000 nm. Since the effective refractive indices of the six modes are faithfully smaller than nwater and larger than nFSM in a wide range of core diameter at each wavelength, the six modes can be well confined within the aqueous core. Since the interest of this work is single-mode properties, we consider the fundamental mode in the following discussion.
3.2 Adiabatic transition condition
The adiabatic transition condition of the tapered MOFs is further investigated. Figure 3(a) shows the confinement losses of the MOFs at different tapering ratios, i.e., with different core diameters, at the wavelengths of 633 nm and 1550 nm. It should be noted that the absorption loss of water is not considered here, and it will be discussed in detail in the discussion part. The confinement losses of MOFs decrease sharply from 3 dB/cm to almost zero within a 200-nm variation of the core diameter, and most of the optofluidic MOFs exhibit low confinement losses. Here, we consider the core diameter that corresponds to a confinement loss of 3 dB/cm in the MOF as the threshold core diameter dad between adiabatic and lossy propagation of light. The simulation shows that the dad is 335 nm at 633 nm, and the confinement loss is calculated as 3.43 dB/cm. And the confinement loss of the MOF is 3.12 dB/cm at 1550 nm when the dad is 855 nm. Both electric field distributions of the MOFs near respective dad at the two wavelengths are presented in the insets of Fig. 3(a), showing weaker confinements of light in the core.
Fig. 3. (a) Confinement losses of the MOFs as functions of the core diameter at 633 nm and 1550 nm. Insets are the electric field distributions of the fundamental mode in the MOFs with confinement losses around 3 dB/cm. (b) Beating lengths of the MOFs with different core diameters at 633 nm and 1550 nm. Insets are the electric field distributions of the fundamental mode in MOFs with minimum beating length: (left) zb of 5 µm; (right) zb of 11 µm.
The limitation of adiabatic mode transition in tapering process is also discussed. It is derived from the adiabaticity criteria that the local taper length should be much longer than the beating length zb between the fundamental mode and the cladding mode or the first higher-order core mode for a negligible power transfer [36]. Then we consider zb as the limitation for the adiabatic transition. Assuming the propagation constants of the fundamental mode and the cladding mode or the first higher-order core mode are β1 and β2, then their beating length zb can be obtained by
Figure 3(b) shows the calculated local beating lengths of the MOFs with different core diameters at the wavelengths of 633 nm and 1550 nm. For the wavelength of 633 nm, when the core diameter increases from 200 nm to 335 nm (the threshold core diameter dad between adiabatic and lossy propagation of light), the beating length zb decreases sharply from 2500 µm to 10 µm. The long beating length comes from the strong coupling between the fundamental mode and the FSM in the cladding, and means a high confinement loss in the MOF. As the core diameter increases to 500 nm (the critical core diameter dsm), the beating length zb decreases slowly from 10 µm to the minimum 5 µm. The short beating length in this range results from the remote beating between the fundamental mode in the core and the FSM. When the core diameter increases further to 5000 nm, the beating length zb climbs to 155 µm. The beating length in this range is longer, due to the close propagation constants of the fundamental mode and the higher-order mode that exists in the core. A similar variation appears at 1550 nm. With the increase of the core diameter, the beating length decreases from 787 µm to the minimum 11 µm, and eventually increases to 68 µm. Both electric field distributions of the MOFs with minimum beating lengths in the respective dsm at the two wavelengths are presented in the insets of Fig. 3(b), showing stronger confinement of light in the core than those in the insets of Fig. 3(a). The short beating lengths presented here indicate that it is not difficult to achieve the adiabatic transition for the MOF tapering process in the subwavelength-scale core range.
4. Effective sensitivity in sensing
In optofluidic sensing applications, it is important to know the fraction of the power in the aqueous core of the proposed MOF. The fraction of power in the core fcore can be obtained by [37]:
Figure 4(a) shows the fraction of power fcore of the water-core MOFs with air-silica claddings at 633 nm and 1550 nm, and for comparison, the fcore of ideal “water-core fibers”, i.e., water-core air-cladding fibers are also presented here. As the core diameter increases from the transmission threshold dad, the curves of fcore grow sharply from around 50% to around 90%, and then climbs slowly to 99.7% and 99.1% in a large core diameter range at 633 nm and 1550 nm, respectively. However, the comparison shows that, the MOFs with core diameters near the dad even exhibit higher fcore values compared with those of the ideal optofluidic “fibers”. As the comparisons shown in Figs. 4(b, c) or Figs. 4(f, g), in these cases, the evanescent fields are dominant in the ideal “fibers” and the electric field distributions are no longer Gaussian-like [28], while the electric field distributions in the MOFs are better confined inside the core. The increasing trends of fcore in the ideal “fibers” are faster than those in our MOFs when the core diameters are larger, however, only the slightly differences are noticed. In detail, for the wavelength of 633 nm, fcore is 58.6% at the core diameter of 400 nm. Additionally, the fcore for the ideal “fiber” with the same core diameter is 61.4%, with a relative difference of 4.6%. As for the wavelength of 1550 nm, the fcore is 68.7% when the core diameter of our MOF is 1100 nm, however, the fcore for the ideal “fiber” with the same core diameter is 69.2%, with a relative difference of 0.7%. In total, the calculation results verify that our designed MOFs preserve excellent performances in confining light inside the aqueous core compared with an ideal condition. The fractions of power at typical core diameters are also investigated. As shown in Fig. 4(a), at the critical diameter dsm, fcore values are about 71.5% (633-nm wavelength) and 78.0% (1550-nm wavelength). Figures 4(d, h) are two electric field distributions of the fundamental modes in the MOFs when the fcore begins to exceed 90% at 633 nm and 1550 nm, respectively. For the wavelength of 633 nm, when the core dimeter increases to 900 nm, the fcore begins to be larger than 90%. A tightly confined fundamental mode pattern is found when the MOF core diameter is 900 nm (Fig. 4(d)). As for the wavelength of 1550 nm, more than 90% of the total power can be confined in the core when the core dimeter is larger than 1900 nm, and Fig. 4(h) is a tightly confined fundamental mode pattern with a MOF core diameter of 1900 nm. As comparison, Figs. 4(e, i) show the electric field distributions of ideal “fibers” when the fcore values begin to exceed 90%, where the core diameter are 675 nm and 1700 nm at the wavelengths of 633 nm and 1550 nm, respectively. The high fraction of power in the aqueous core of our designed MOFs is exceedingly beneficial to enhancing light-fluidic interactions in the aqueous cores, and shows great potential in optofluidic applications.
Fig. 4. (a) Fraction of power in the aqueous core as functions of the core diameter in the MOF in the cases of air-silica cladding and air cladding at 633 nm and 1550 nm. (b-i) Electric field distributions of the fundamental modes in aqueous-core MOFs under the conditions of: core diameter = 335 nm, (b) fcore=45.6%, (c) fcore = 43.3%; (d) core diameter = 900 nm, fcore = 90.7%; (e) core diameter = 675 nm, fcore = 90.4%; core diameter = 855 nm, (f) fcore = 48.1%, (g) fcore = 45.3%; (h) core diameter = 1900nm, fcore = 90.8%; (i) core diameter = 1700nm, fcore = 90.4%. (j) Sensitivity coefficient of the MOF as functions of the core diameter at 633 nm and 1550 nm.
The relative sensitivity coefficient r is essential in characterizing the performance of the absorption-based fluidic sensing, which can be defined as [38]
Figure 4(j) shows the calculated sensitivity coefficient r of the MOFs at 633 nm and 1550 nm. When the core diameter increases from the core diameter dad, the value of r increases sharply to 99.9% that benefitted from the enhanced overlap between light and aqueous solution. It is worth noting that the minimum sensitivities at each core diameter dad are 51.6% and 57.0% at the wavelengths of 633 nm and 1550 nm, respectively. The enhanced sensitivity presented here are much larger than those of the reported D-shaped fiber, solid-core MOF, and the microstructured-core MOF that rely on the interaction between the evanescent field and fluidic samples [37,38], serving as a promising paradigm for the optical sensing.
5. Enhanced nonlinearity
5.1 Effective mode area
The effective mode area Aeff of an optical fiber is originated from the measure of nonlinearity, and is defined by [18,39]
We calculated the Aeff of the fundamental modes in the MOFs as functions of the core diameter at the wavelengths of 633 nm and 1550 nm. As shown in Fig. 5, since the amount of power in the evanescent field increases at smaller subwavelength core diameters, Aeff first drops when the core diameter ranges from 200 nm to 400 nm. And then Aeff raises as the core increases. The smallest Aeff is 0.38 µm2 when the core diameter dAeff is 400 nm at 633 nm, which represents the core diameter with the optimal mode confinement and is around 31-fold smaller than that in the initial 5 µm core MOF. For the wavelength of 1550 nm, Aeff has a similar variation trend. The smallest Aeff is 2.2 µm2 when the corresponding core diameter dAeff is 1100 nm and it is around 6-fold smaller than that in the initial 5 µm core MOF. It is worth noting that both minimum Aeff at each wavelength locate in their single mode regions, and the ultra-small Aeff at each wavelength suggests a great potential to enhance the power density in the subwavelength-scale core.
Fig. 5. (a) Effective mode area of the fundamental mode as functions of the core diameter in the optofluidic MOF at 633 nm and 1550 nm. (b) Power density of the fundamental mode as functions of the core diameter in the MOF at 633 nm and 1550 nm.
5.2 Power density
An intense optical field is important for nonlinear applications. Therefore, we calculated the power density D of the fundamental mode in the core of MOFs to evaluate the capability of these MOFs in nonlinear applications. Here we define D in the optofluidic MOF as
where P is the power carried by the fundamental mode and is assumed to be 1 W for all calculations, fcore is the fraction of power in the core, and Score is the area of the core.Figure 6(b) shows the calculated power density D of the fundamental mode in the core of these optofluidic MOFs at two wavelengths. The figures show that D increases firstly and then decreases as the core diameter increases from 200 nm to 5000 nm. More specifically, for the wavelength of 633 nm, D increases from 0.047 W/µm2 to 4.7 W/µm2 when the MOF is changed from the initial core diameter 5 µm to 335 nm, showing a large intensity enhancement factor of 100. At the wavelength of 1550 nm, D exhibits a 17-fold increase from 0.046 W/µm2 to 0.77 W/µm2 when the MOF is changed from the initial core diameter to 855 nm. The tapered structure enhances the intensity efficiently, indicating the potential in effective manipulation of optical property with low power.
Fig. 6. (a) Effective refractive indices of the fundamental modes in MOFs with different core diameters of 0.85 µm, 1 µm, 1.5 µm, and 2 µm as functions of the wavelength. (b) GVD curves of MOFs with different core diameters of 0.85 µm, 1 µm, 1.5 µm, and 2 µm as functions of the wavelength. The light gray dash line represents the zero dispersion.
5.3 Nonlinear coefficient
Many nonlinear applications of the MOF, such as Stimulated Raman Scattering and four wave mixing are related to the nonlinear coefficient γ. The definition is given by [16,39]
where n2 is the nonlinear refractive index of the material and λ is the optical wavelength. As expected, γ can be enlarged by choosing the material with a high value of n2, or decreasing the effective mode area Aeff of the fiber.Then we compare the improvement of nonlinear coefficients γ of the proposed initial and tapered MOF. As we have mentioned in the effective mode area section, compared with the initial MOF, Aeff values of the tapered MOF are 31-fold and 6-fold smaller at 633 nm and 1550 nm, respectively. Thus, γ performs an increase of the identical 31-fold and 6-fold factors in the tapered optofluidic MOFs respectively.
The nonlinear coefficients of the proposed MOF and a commercially available hollow-core photonic bandgap fiber (HC-PBGF) HC-1550-02 from NKT Photonics are also compared. Since HC-1550-02 fiber can guide light in the aqueous core [40], we calculated the Aeff of the water-core HC-1550-02 fiber. The Aeff values are 53.37 µm2 and 53.46 µm2 for the wavelengths of 633 nm and 1550 nm, respectively. And the minimum Aeff values 0.38 µm2 and 2.2 µm2 of the tapered MOFs are around 140-fold and 24-fold smaller than those of HC-1550-02 fiber at the wavelengths of 633 nm and 1550 nm, implying the identical-fold increases of γ. The liquids such as CS2, toluene, ethanol, etc [19,41,42]. may be suitable for nonlinear optofluidic applications using our tapered MOF due to the high nonlinearity of liquids. Since different liquids have different refractive indices and material dispersion properties, core diameters should be carefully chosen in the waveguide design in terms of zero dispersion wavelength and effective mode area. For instance, to implement stimulated Raman scattering of the ethanol-core MOF [41], the optimum core diameter can be around 320 nm with the minimum effective mode area of 0.25 µm2 at the wavelength of 532 nm, which will dramatically improve the efficiency of stimulated Raman scattering generation. These comparisons show that the large nonlinear coefficient of the tapered MOF will greatly reduce the required power and facilitate the exploration of nonlinear effects [12,39,43].
5.4 Group velocity dispersion
Tailoring the physical parameters of optical fibers can generate tunable GVD values at the desired wavelength [12], which is important for the optical communication and nonlinear effects, since the zero-dispersion point allows for extended effective interaction lengths [14,44,45]. Thus, we calculated the effective refractive indices, and diameter- and wavelength- dependent GVD values of the tapered MOFs to explore the potential in applications. The GVD at frequency ω can be obtained from the propagation constant β(ω) of the simulation model [45]:
The effective refractive indices of the fundamental modes in MOFs with different core diameters of 0.85 µm, 1 µm, 1.5 µm, and 2 µm are shown in Fig. 6(a). As the wavelength increases from 300–2000 nm, both the material dispersion property and the leakage of the fundamental mode into the cladding contribute to the decrease of neff. Besides, when the core diameter d grows from 0.85 µm to 2 µm, light can be better confined in the core.
Figure 6(b) shows the calculated GVD properties of MOFs with different core diameters of 0.85 µm, 1 µm, 1.5 µm, and 2 µm. For example, large anomalous GVD lies in a wide portion of wavelength ranging 300–2000 nm when d is 0.85 µm or 1 µm, while the normal GVD with large variations can also be observed when d is 2 µm. Besides, the GVD curve with flat variations and small values are presented when d is 1.5 µm. It is worth noting that the GVD curves show blue-shifts as d decreases, so that the zero-dispersion wavelength can be tuned over a broad spectral range. For example, when d decreases from 2 µm to 0.85 µm, the first zero-dispersion wavelength blue-shifts from 717 nm to 579 nm in the visible spectral range. In the near-infrared region, the second zero-dispersion wavelength blue-shifts from 1920 nm to 950 nm when d decreases from 1.5 µm to 0.85 µm. The results indicate that we can tune the dispersion properties flexibly in a wide spectral range covering from visible to near-infrared region by changing the core size, which is important for generating supercontinuum. Both the ultra-small Aeff and controllable dispersive properties of the tapered MOF are beneficial to the high efficiency of nonlinear processes.
6. Discussion
We have investigated the properties of subwavelength-scale optofluidic MOFs and demonstrated the potential advantages. However, several issues such as sample consumption volume, infiltration duration and fabrication, interface coupling and splicing, and fluidic material attenuation should be considered for the practical utilization.
6.1 Sample consumption volume
The estimation of sample volumes is based on a 3-cm fiber length for the proposed MOF structure. For the aqueous-core MOF with a core diameter of 5 µm, Score is calculated as 21.43 µm2 and the sample volume is around 0.64 nL. However, for the tapered aqueous-core MOFs with core diameters of 335 nm and 855 nm, Score are 0.096 µm2 and 0.63 µm2, the sample volumes are around 2.88 fL and 18.9 fL, respectively. The subwavelength structure reduces the sample volume by 4–5 orders of magnitude, which may be considerably beneficial for the ultra-trace sample detection.
6.2 Fluidic sample infiltration duration and fabrication issues
The qualitatively prediction on the infiltration duration of water into the central hole of the MOF is based on the filling model in photonic crystal fibers [46]. Since the central hole diameters of the MOFs are less than 5 µm, the gravitational force is negligible [46]. Besides, we consider no pressure is applied to the liquid end. Thus, only the capillary force and the friction force in total act on the water column inside the hole. Due to the small size of the central hole in the MOF, the water flow inside the hole will be laminar, and we can solve the corresponding differential equation [46] to determine the infiltration durations in MOFs. The calculated infiltration durations for 3-cm-long MOFs with core diameters of 335 nm, 855 nm, and 5 µm are 75 s, 29 s, and 5 s, respectively. The flow rate of 732 µm/s in the hole of 600 nm can be obtained when the filling pressure is 5 bar [46]. The reasonable infiltration speeds indicate the small hole will not prevent the potential utilization of subwavelength-scale aqueous-core MOF. Selectively filling process is critical in the practical fabrication, and there are various selective infiltration methods [21,40,47]. Since the micro-holes at the end-face of MOF are uniform size, the two-photon direct laser writing technique [21] or the microfiber glue-sealing method [47] will be preferred to use, both of them have successfully demonstrated the flexibility of selective infiltration. To avoid the possible hole collapse during tapering with a large taper ratio, a “fast and cold” tapering process or a hole-pressurization tapering process [48] may be utilized in the fiber post-processing.
6.3 Interface coupling and splicing issues
The tapering structure of our MOF will facilitate the optical coupling to the subwavelength-scale optofluidic fiber waveguide. Moreover, at the interface between the optofluidic-core MOF and the input fiber, the mode-matching should also be considered. The small-core single mode fiber with high numerical aperture, such as Nufern UHNA fiber will be a good candidate fiber to act as the input fiber, which is in general used as the input fiber for high-index contrast waveguides. Since the liquid-core fiber could not bear high-temperature in the normal fusion splicing process, the mechanical splicing would be preferred in the practical operation [19].
6.4 Attenuation induced by water absorption
It is well known that there is a prominent water absorption band in the near infrared spectral range at around 1450 nm [49], and the wavelength of 1550 nm is disturbed more or less. As a result, despite the inherent low confinement loss of the aqueous-core MOF, the length of the fiber should be considered as an important factor that affects the transmission efficiency due to the absorption of water. The absorption coefficient of pure water at 20°C is around 10 cm−1 at the wavelength of 1550 nm [49], i.e., for a 3-cm-long water-core MOF, the attenuation of power due to the absorption of water is around 120 dB. However, the absorption coefficient of water at 20°C is 0.003 cm−1 at 633 nm [50], and the corresponding attenuation of power is only 0.03 dB for a 3-cm-long water-core MOF. Therefore, the material attenuation of microfluid in different spectra range should also be considered.
7. Conclusion
We have proposed a tapered hollow-core MOF structure with both light and water confined inside its central core, and investigated the performance of the MOF which has a core diameter in the subwavelength scale using the full vector FEM. The basic optical characteristics of the subwavelength-scale optofluidic MOF are studied for the first time, including the modal guidance condition and the adiabatic transition condition. The limitation of sensing sensitivity is explored by the studies on the power fraction in the core and the relative sensitivity coefficient, while the nonlinearity in the sub-wavelength scale MOF waveguide is elaborated in terms of the effective mode area, the power density, the nonlinear coefficient, and the group velocity dispersion. Besides, the sample consumption volume, the fluidic sample infiltration duration and fabrication issues, the interface coupling and splicing issues, and the attenuation induced by the absorption of water are discussed to provide a guidance for utilization. The detailed analysis on the subwavelength-scale optofluidic MOF structure shows an ultra-high sensitivity in sensing and enhanced nonlinearity, with the capability in device fabrication, which will pave the way for the applications of the subwavelength-scale optical fiber devices in optical sensing and nonlinear optics.
Funding
National Natural Science Foundation of China (61775041).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be available from the corresponding author upon reasonable request.
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