Fiber Bragg grating (FBG) has delivered outstanding performance for applications in many fields of fiber-optic sensors, lasers, dispersion management, modulators, and filters, and has arrived at a high maturity level.^[1–5] Recently the femtosecond-laser inscription technique^[6,7] has effectively revolutionized the FBGs fabrication, and extended the applications to different types of optical fibers, as shown in Table 1.

Table 1.

Table 1.

Table 1. State-of-the-art of FBGs fabricated in different optical fibers. . Ref. Optical fibers Applications Cores [1] step-index fiber various GeO2-doped, solid [2] step-index fiber sensor GeO2-doped, solid [3] HB step-index fiber sensor GeO2-doped, solid [4] step-index fiber sensor GeO2-doped, solid [5] step-index fiber sensor GeO2-doped, solid [6] step-index fiber various GeO2-doped, solid [7] index-guiding PCF various GeO2-doped, solid [8] multicore fiber sensor GeO2-doped, solid [9] multicore fiber filter GeO2-doped, solid [10] coated core fiber sensor GeO2-doped, solid [11] all-solid PBG MOF sensor GeO2-doped, solid [12] air-hole cladding MOF sensor GeO2-doped, solid [13] metallic-hole MOF sensor GeO2-doped, solid [14] small-hole MOF sensor GeO2-doped, solid [15] multi-core PCF sensor pure silica, solid [16] suspended-core MOF filter pure silica, solid [17] suspended-core MOF modulator pure silica, solid [18] suspended-core MOF sensor pure silica, solid [19] suspended-core MOF sensor pure silica, solid [20] random clad MOF sensor pure silica, solid [21] all-solid Bragg MOF sensor GeO2-doped & pure silica, solid [22] nanostructure core fiber sensor GeO2-doped rods array, solid [23] dual-core MOF sensor hydrogen loaded silica, solid [24] hollow eccentric fiber sensor hydrogen loaded silica, solid [25] exposed core MOF sensor silica, solid with drilled holes [26] micro-fiber various silica, solid with drilledc holes This work nano-bore fiber various GeO2-doped, with nano-bore

Table 1. State-of-the-art of FBGs fabricated in different optical fibers. .

The FBGs have been successfully inscribed in the index-guiding photonic crystal fiber (PCF) with the GeO₂-doped solid core^[7] by the femtosecond-laser, and are also inscribed in the multi-core fibers for applications in sensors^[8] and filters.^[9] A novel CO₂ sensor at room temperature was proposed by using the FBG fabricated in the photo-sensitive optical fiber with the carbon nanotubes coated core.^[10] A dual-peak FBG was inscribed in an all-solid photonic bandgap (PBG) fiber for sensing applications.^[11] Furthermore, for the micro-structured optical fibers (MOFs) with solid core, FBGs have also been fabricated in the silica-polymer,^[12] metallic-hole,^[13] and small-hole^[14,15] MOFs for applications in high-performance sensing.

Besides FBG written in the GeO₂-doped optical fibers, the high power ultraviolet and femtosecond laser have also been successfully applied to fabricating FBGs in the pure silica optical fibers. The FBGs can be inscribed in the suspended-core MOFs for applications in filters,^[16] modulators,^[17] and sensors.^[18,19] Besides the suspended-core MOFs, FBG can also be fabricated in the pure silica solid core of MOFs with the random air-line cladding for structural health monitoring within nuclear reactors.^[20] For the all-solid Bragg MOFs, the FBG can also be written in its GeO₂-doped and pure silica solid core by femtosecond laser for sensing.^[21] The FBGs have been inscribed in the solid core with GeO₂-doped rods array in a nanostructure core fiber,^[22] in the solid hydrogen loaded silica core of dual-core MOF,^[23] and in the hollow eccentric fiber.^[24] Besides the photosensitivity, the femtosecond laser ablation drilled periodic nano-holes can also be utilized in the exposed core MOFs and microfibers.^[25,26]

Currently, for both the conventional step-index optical fibers and various different MOFs, FBGs are all written in the solid cores.^[1–26] It is obvious that the FBGs cannot be fabricated in the hollow core (HC) of optical fiber. Therefore, there is little work on the attempt at combining the technical functions and advantages of both solid-core FBGs and hollow-core optical fibers (HCFs).

However, the lately proposed special optical fibers, namely, the nano-bore optical fibers (NBFs)^[27–30] may produce the possibility to make a breakthrough. The NBF consists of a GeO₂-doped core, a silica cladding, and a central hollow nano-bore in the core.^[27–30] Several NBFs have been reported as shown in Table 2. The NBFs with step-index and GeO₂-doped cores have been proposed with air,^[27] liquid,^[27,28] dielectric particles,^[29] and gold nano-wire^[30] in the nano-bore for applications in sensing,^[27] spectroscopy,^[28] nanoparticle tracking,^[29] plasmonics, and optics.^[30] Even a nano-bore with a 20-nm diameter was successfully fabricated in the solid core of a soft glass MOFs for sensing and nonlinear optics.^[31] What is more, the silica-chalcogenide NBFs with thermo-optic liquid filled single^[32] and dual^[33] nano-bores have also been numerically and experimentally demonstrated for applications in parametric sources, light–matter interaction, and spatial soliton propagation.

Table 2.

Table 2.

Table 2. Current work on NBFs. . Ref. Optical fibers Cores Dbore/nm Filling in bores Applications [27] step-index fiber GeO2-doped 180 (∼940 air and liquids sensors [28] step-index fiber GeO2-doped 600 liquid spectroscopy [29] step-index fiber GeO2-doped 247 ± 6 dielectric particles nanoparticle tracking [30] step-index fiber GeO2-doped 1100 gold nano-wire Plasmonics & Optics [31] soft glass MOF soft glass 20 (∼150 air sensing & nonlinearity [32] silica-chalcogenide chalcogenide 200 (∼1600 thermo-optic liquid parametric sources [33] silica-chalcogenide chalcogenide 500 (∼1000 thermo-optic liquid light-matter interaction

Table 2. Current work on NBFs. .

Current work on NBFs focuses on taking full advantages of selective filling and maintaining total internal reflection in the core with the nano-bore. However, the NBFs make possible the hollow and filled FBGs due to the presence of both solid GeO₂-doped core and the hollow nano-bore. In this paper, hollow and filled FBGs in the NBFs are proposed and investigated in detail to attempt to integrate the FBGs and HCFs together.

The NBF studied here is based on a silica step-index optical fiber, and consists of a GeO₂-doped silica core, and a silica cladding, and there is a central hollow nano-channel extending through the entire length of the NBF as shown in Fig. 1.

Fig. 1.

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	Fig. 1. Sketch of NBF consisting of a GeO2-doped core, silica cladding, central hollow nano-channel,[27–30] and FBG in NBF.

The diameters of the cladding and the core of NBF are and , respectively. The diameter of the nano-bore D_bore changes from 50 nm to in our numerical investigations. The refractive index of the silica cladding is n_cladding=1.45 for 1550-nm wavelength. The GeO₂-doped concentration in the core is 5.3 mol%, then, the refractive index difference between core and cladding is 8×10⁻³, therefore, the refractive index of the core is n_core=1.458. The refractive index of the hollow nano-bore is n_bore=1. It is worth noticing that the high Ge-content can offer intrinsic photosensitivity for FBG inscription without hydrogenation.

Here, we analyze several application-relevant properties of the NBFs by the full-vector finite element method (FEM),^[34] including the fundamental mode field, the confinement loss, and effective mode index of the fundamental mode with different nano-bore diameters. The obtained results will pave the way for fabrication of FBGs in the NBFs.

The fundamental mode fields of NBFs with different nano-bore diameters when the wavelength is 1550 nm and n_bore=1 are shown in Fig. 2. The results are all for 1550-nm wavelength and a hollow nano-bore (n_bore=1). The values of effective index n_eff of the fundamental mode with different values of bore diameter D_bore are illustrated in Fig. 3. When D_bore increases, n_eff decreases due to the decrease of the proportion of silica in the core.

Fig. 2.

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	Fig. 2. Images for normalized electric field intensity of fundamental mode in NBF with different bore diameters.

Fig. 3.

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	Fig. 3. Variation of effective index of fundamental mode (neff) with bore diameter (Dbore).

Then, the confinement loss of fundamental mode ((α) as a function of D_bore is illustrated in Fig. 4. For the NBF with hollow bore, when , thers is no obvious confinement loss, however, (α increases remarkably from 5.85 dB/m to 10.62 dB/m when .

Fig. 4.

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	Fig. 4. Plot of confinement loss ((α) of fundamental mode versus nano-bore diameter (Dbore).

Then the cross section of the normalized intensity of the mode field is shown in Fig. 5. In order to quantitatively describe the power in the bore, an important parameter, power fraction (η), is used, which represents the ratio of the power inside the nano-bore to the total guided power of the fundamental mode, and η can be obtained by integrating the z-component of Poynting vector (S_z) in the bore (A_bore) and the core (A_core) domains, as follows:

Fig. 5.

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	Fig. 5. Plots of cross section of normalized mode field intensity versus position in x direction for different values of bore diameter (Dbore).

Fig. 6.

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	Fig. 6. Plot of power fraction (η) of NBFs (8- core diameter) with hollow bore versus bore diameter (Dbore).

The power fraction η lies in a range between 0.0047% and 1.2101% when D_bore increases from 50 nm to 900 nm, suggesting that the influence of the lossis negligible as long as fiber length below about 10 cm is considered. When D_bore is in a range of , the power fraction η observably increases from 1.3071% to 9.3734%. Furthermore, there is a high η value of 53.5030% in the case of 7- D_bore. It is necessary to note that they are very special cases where and in Fig. 5. The most of the intensity is outside the core, which leads to a very low reflectivity of FBG as shown in Fig. 7 (see below). There is very little light in the bore, and also in the core, therefore, there is a relatively high η when D_bore is larger than .

Fig. 7.

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	Fig. 7. Reflected spectra of hollow FBG in NBF for different bore diameters.

Larger bore can lead to the high power fraction, which is useful for enhancing the light–matter interation, the sensitivity of the absorption spectrum and chemical sensing, and the power fraction can be enhanced to (∼30% by arranging the geometric parameter and selective filling.^[35] However, the larger D_bore also brings in the high confinement loss, and the characteristic of FBG in NBF may be degenerated, which will be studied in the next section.

The FBG can be inscribed in the GeO₂-doped and solid part of NBF core as shown in Fig. 1. Generally, a uniform FBG can be described by a modulation of the effective refractive index (n(z)) of the fundamental guided mode along the fiber axis z. The n(z) is given as follows:

With the purpose of carrying out the numerical analysis of the FBG in NBF, the coupled mode theory based transfer matrix method (TMM)^[1] is utilized by taking into account the existence of the nano-bore. The FBG is based on the photosensitivity, which is described as the light-induced permanent effective refractive index change. In most of cases, the photosensitivity of single-mode fiber is initiated by the breaking of Ge–Si bonds with UV radiation,^[36] which is strongly supported by hydrogen loading.^[37] A general value of for simulation is .^[38] However, for the FBG in NBF, the index is reduced due to the nano-bore in the core, and is approximately estimated in our simulatiom from

Then, the reflected spectra of FBG in NBF with different bore diameters are shown in Fig. 7. Furthermore, to observe the spectra more quantitatively, the maximum reflectivity (R_max) and the Bragg wavelength ( ) with different values of D_bore are shown in Figs. 8 and 9, respectively. It is found that for D_bore in a range from 50 nm to , R_max keeps more than 85%, however, when , R_max descreases quickly; even for , R_max is 0, which means that there is no available FBG in NBF. The reason is that when D_bore increases, the confinement loss is enhanced, meanwhile, the solid part of the core is also reduced, which remarkably weakens the index changing for forming the FBG and also the diffraction by FBG.

From Fig. 9, it is found that shows similar dependence on D_bore to n_eff as shown in Fig. 3, due to the quantitative relationship between and effective index of FBG^[1]

Fig. 8.

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	Fig. 8. Variation of maximum reflectivity (Rmax)versus bore diameter (Dbore) of FBG.

Fig. 9.

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	Fig. 9. Variation of Bragg wavelength ( ) of FBG with bore diameter (Dbore).

It is found that the hollow FBG in NBFs can be realized with (∼80% reflectivity, and the hollow bore for selective filling will be studied in the next section in detail.

The filled FBG in NBF can be realized by selective filling in the bore. We choose three different fillings including o-xylene, trichloroethylene, and chloroform^[39] with different values of effective index (n_filling), which are less and greater than the index of core (1.458). It is noted that the fluctuation of n_filling with the change of wavelength is ignored.

The effective mode index (n_eff) and power fraction (η) of the filled NBF, the maximal reflectivity (R_max), and Bragg wavelength ( ) of filled FBG are shown in Table 3. The material induced attenuation of filling is ignored.

Table 3.

Table 3.

Table 3. Filled FBGs in NBFs with different fillings and Dbore values. . Filling nfilling Dbore/nm neff η/% Rmax /nm o-xylene 1.4846 300 1.4544 0.1733 0.9982 1559.2 800 1.4549 1.3461 0.9865 1559.7 Trichloroethylene 1.4601 300 1.4543 0.1716 0.9982 1559.1 800 1.4544 1.2262 0.9877 1559.1 Chloroform 1.4321 300 1.4543 0.1698 0.9982 1559.0 800 1.4540 1.1555 0.9884 1558.7

Table 3. Filled FBGs in NBFs with different fillings and Dbore values. .

The effective mode index n_eff increases when D_bore changes from 300 nm to 800 nm for o-xylene and trichloroethylene with the index n_filling greater than core index. By contraries, n_eff decreases with larger D_bore for chloroform with n_filling less than core index. The power fraction η for D_bore=800 nm is 10 times larger that for D_bore=300 nm. The reflected spectra of filled FBGs and fundamental mode field of NBFs with different bore diameters and fillings are illustrated in Fig. 10. It is obvious that the light confinement is enhanced by both the increased n_eff, and the larger D_bore for the xylene and trichloroethylene due to their greater n_filling. In the cases of these three fillings with 300-nm and 800-nm bore diameters, R_max maintains greater than 0.98.

Fig. 10.

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	Fig. 10. Reflected spectra of filled FBG and fundamental mode field of NBF with different bore diameters and fillings.

In the case of the o-xylene, trichloroethylene, and chloroform filled NBF with 800-nm D_bore, the power fraction η can reach (∼1%. This value is effective for absorption spectrum to detect high-concentration analytes at experimentally feasible length, which is hard for current cuvette-based experiments. For a power fraction factor of 0.1%, a 1-cm-long NBF allows measuring a 1000-times higher analyte concentration than a 1-cm-long cuvette.^[28] What is more, it is interesting to find that the bore with 1% η can be utilized in absorption spectroscopy and chemical sensing, and the GeO₂-doped and solid part of core leads to the FBGs with high reflectivity, which means that a filled FBG can be realized.

In order to analyze the feasibility of the index sensing by the NBF FBG, the equivalent wavelength sensitivity of the Bragg wavelength to the filled index can be calculated from

Table 4.

Table 4.

Table 4. Equivalent sensitivities with different bore diameters. RIU: refractive index unit. . Dbore/nm) /(nm/RIU*) 300 (1559.2–1559.0)/(1.4846–1.4321)= 3.8095 800 (1559.7–1559.0)/(1.4846–1.4321)= 13.3333

Table 4. Equivalent sensitivities with different bore diameters. RIU: refractive index unit. .

The NBF can be fabricated by thermally drawning the centimeter-thick fused silica and preformed with a GeO₂-doped core and a hollow central channel at high temperatures into a fiber.^[27–30] Then, the uniform FBGs in NBF can be fabricated by the phase-mask scanning technique,^[38,40] where the 248-nm UV radiation from a frequency-doubled argon laser is focused by a cylindrical lens through a uniform phase mask onto the core of NBF. However, the existence of the hollow bore in the core will strongly defocus the UV beams, especially for the large bores. We believe that it may be an effective solution to fill the bore with the suitable material with high index nearly the same as the index of the GeO₂-doped core and low loss for UV beams during the exposure progress. Furthermore, the power of the UV laser and the scanning speed need optimizing for different bore diameters for the cases with or without the hydrogen-loading. In order to improve the long-term reliability of FBG in NBF, the annealing may also need implementing for practical applications. These engineering issues will be studied in our future work.

For applications in sensing, the chemical patameters such as solution concentration, relative humidity, and hydrogen concentration can be measured by employing our proposed NBF FBG as a refractive index sensor by filling the analyte in the hollow and selective coating (hydrogel humidity for and palladium for hydrogen) bores. The physical parameters including strain and temperature can also be detected by our proposed FBG in NBF.

In this paper, we report the detailed theoretical investigations of hollow and filled FBGs in the NBFs, which is an integration of HCFs and FBGs. The GeO₂-doped solid part of the core in NBF is employed to fabricate FBG, and the nano-bore of the NBF core can be selectively filled, to form a hollow or filled FBG. By the full-vector FEM, the transmission characteristics including the fundamental mode field, effective mode index, and confinement loss of NBF with different-diameter hollow and filled nano-bore are numerically investigated. Then, the reflected spectra of FBGs in the NBFs are obtained by the TMM, based on the numerical results of NBF transmission characteristics. The range of bore diameter for available power fraction in the bore and the range of FBG reflectivity are obtained. For the hollow FBGs, the (∼5% power fraction in the bore and the (∼0.9 reflectivity can be realized with bore diameter. For the filled FBGs, the (∼1% power fraction and 0.98 reflectivity with different fillings including o-xylene, trichloroethylene, and chloroform can be achieved by using an 800-nm bore diameter. The feasibility of the index sensing by our proposed NBF FBG is analyzed and discussed. We believe that this work is a useful reference for the integrated fiber-optic platforms with HCFs and FBGs.