An alternative way is to use a 248 nm KrF excimer laser and a uniform phase mask to inscribe FBG in MFs drawn from 125 μm-diameter fibers [26,27]. The preform fibers are usually highly Ge-doped and have large cores to guarantee that the MFs have a large enough photosensitive cross section. Sometimes, hydrogen loading treatment are further employed to increase the photosensitivity [27]. However, during the hydrogen loading treatment, high pressures and temperatures are needed, which complicates the procedure [27].

To avoid additional hydrogen loading or photosensitization treatments, Ran et al. used a 193 nm ArF excimer laser and phase mask to inscribe strong FBG in MFs drawn from both standard telecom SMF [28] and 62.5/125 μm multimode fiber (MMF) [29]. This method utilizes the high efficiency associated with two-photon excitation at 193 nm [30].

FIB milling, a powerful micromachining technique, has also been used to fabricate MFBGs [31–38]. This method employs accelerated ions to mill nanometer-scale features on MF surfaces to form corrugated structures. As the index modulation results from changes in the structure, this kind of gratings are called structural MFBGs (sMFBGs).

Prior to the milling, the MF is coated with a thin film of metal, e.g., aluminum or gold [31–35], to prevent charge accumulation which cause ion deflections and large fabrication errors. Alternatively, MF can also be laid on a doped silicon wafer [36]: due to van der Waals' forces, the MF tightly attaches to the conductive substrate and it avoids charging by transferring charges to the wafer.

During the FIB micromachining process, the MF sample should be fixed firmly in the vacuum chamber to minimize sample displacements. A 30 kV, 10–300 pA Ga⁺ ion beam is usually used to get enough milling accuracy depending on different FIB systems. The total milling process takes minutes to hours according to the beam current used and milling area. After the machining process, for a nonmetallic sMFBG, the MF is immersed in metal etchant to totally remove the metal film and then is cleaned with deionized water.

Figure 2 shows FIB/SEM pictures of sMFBGs fabricated from different groups. Gratings in Figure 2(a–c) are fabricated on MF tips while the rest are on tapers. The sMFBGs are fabricated on MFs with diameter ranging from 560 nm (Figure 2(f)) to 6.6 μm (Figure 2(a)) and the number of period of the gratings varies from 11 (Figure 2(c)) to 900 (Figure 2(e)). Both high (5 × 10⁻³−10⁻¹, Figure 2(c,d,f)) and low (10⁻⁴−5 × 10⁻³, Figure 2(a,b,e)) average RI modulations have been achieved by FIB-milled sMFBG. In all, FIB provides researchers a flexible way to get all kinds of structures with high accuracy at will and without additional masks. Yet, batch production cannot be envisaged for this method.

Femtosecond-laser-irradiation is another way to cause periodically physical deformation on the surface of MFs [40]. During the femtosecond laser irradiation, the ultra-short laser pulse transfers energy to the electrons in the material irradiated through nonlinear ionization [45]. When a sufficiently high energy is achieved, pressure or shock wave will cause melting or non-thermal ionic motion, resulting in permanent structural damages in the material. Aided with proper phase masks, MFBGs can be fabricated on the surface of the MFs [39,40].

In addition to the previous techniques, other methods have also been demonstrated or proposed. MFBGs can be manufactured by wrapping a MF on a microstructured rod with an internal channel (see Figure 3(a)) or by laying the MF on a substrate with pre-treated microstructures (see Figure 3(b)). By exploiting the fraction of power propagating in the periodically distributed patterns in the rod or the substrate, light transmission could be modulated. This compact scheme of Figure 3(a) can be used as a RI sensor when the evanescent field extends in the inner fluidic channel. Both methods avoid post-processing the thin MFs and have great flexibility. However, the MFs have to be coated with low index polymer [46] which means that they are not suitable for high temperature sensing.

Ding et al. [42] combined metal lift-off technology with lithography to produce metallic surface gratings, which provided a high and constant sensitivity to the ambient medium RI, while Phan Huy et al. [43] demonstrated an improvement in the sensitivity of RI by making use of the suspended core of a microstructured fiber.

In the grating, the forward (f) and backward (b) propagating modes are related by (1) β b → = β f → + m 2 π Λ j → .where β_i = (2π/λ)n_eff,i (i = f or b) is the mode propagation constant, m is the diffraction order, Λ is the period of the grating and J⃗ is the unit vector along z-axis, as illustrated in Figure 1. In a more physical perspective, Equation (1) means the momentum mismatch between the forward and backward propagating modes should be compensated by the reciprocal vector provided by the periodical index modulation. For the first-order diffraction which is commonly seen in MFBGs: (2) ( n eff , f + n eff , b ) Λ = λ B .

Furthermore, if the two modes are identical, the commonly used Bragg resonance condition can be obtained, namely, λ_B = 2n_effΛ.

Before designing the grating, n_eff should be calculated first. For a uMFBG, the index modulation is relatively weak (usually on the order of 10⁻⁴) and the cross section of the MF is a symmetrical circle. n_eff can be easily obtained solving the dispersion equations numerically.

However, for a sMFBG, the effective index difference between the MF milled and un-milled cross section can be as large as ∼10⁻³ [31], or even ∼10⁻¹ [34], orders of magnitude larger than that in conventional FBGs. One way to calculate an averaged effective index of the grating region is to choose an unperturbed waveguide boundary [48], shown as the curved dashed line in Figure 4(b). d is the depth of the corrugation and h_eff is the boundary shift from the top of the corrugation to the new boundary of the corresponding unperturbed waveguide. The boundary shift h_eff as illustrated in Figure 4(b,c) is determined such that the volume bounded by the upper part of the corrugation (S_Aτ) is equal to that of the volume bounded below [S_B(1−τ)], i.e., (3) S A τ = S B ( 1 − τ ) .where τ is the duty cycle. By assuming that τ = 0.5 and d ≪ r_MF, we get h_eff ≈ d/2. The averaged effective index could thus be obtained by mode analysis after the unperturbed waveguide boundary is established.

The uMFBGs reflection spectrum can be estimated using [49] (4) R = sinh 2 ( κ 2 − γ 2 L ) cosh 2 ( κ 2 − γ 2 L ) − γ 2 κ 2 .where κ, γ, L is the coupling coefficient, self-coupling coefficient, and length of the grating, respectively. Due to the weak index modulation seen in the uMFBGs, MFs with grating region up to several hundreds of micrometers or millimeters in length is needed in order to get detectable reflection [22,23,26].

However, due to the large index difference in the corrugation region in sMFBGs, strong scattering may occur. A more effective way to verify the experimental spectrum obtained from the sMFBG is a 3D finite element simulation, shown in Figure 5. This method takes the details of the structural deformation into consideration and thus better reflects the real situation experienced by the light field.

Most MFBG sensors rely on the monitoring of the shift in wavelength of the reflected Bragg signal with the changes in the measurand (e.g., refractive index, temperature, and strain/force).

As the effective index and period of grating is a function of r_MF, n_a, T, and ε, the Bragg condition can be rewritten as (5) λ B = 2 n eff ( r M F , n f , n a , T , ε ) Λ ( T , ε ) .where the RI of the fused silica and the ambient medium surrounding the MFBG are denoted by n_f and n_a, T is the operating temperature and ε is the strain applied to the MFBG.

Much of the MFBG applications relate to RI sensing. For a typical MFBG sensor immersed in ambient liquid with RI in the range 1.32–1.46, S_a varies from 10¹ nm/RIU (refractive index unit) to 10³ nm/RIU, according to the MF radius and the ambient liquid sensed. Usually, a smaller radius and a larger ambient medium RI result in a higher sensitivity regardless of the fabrication method. For example, Liang et al. got a sensitivity of 16 nm/RIU at a RI around 1.35 with a MF 6 μm in diameter [22] while 660 nm/RIU was reached by Liu et al. at a RI of 1.39 by using a 1.8 μm-diameter MF [36]. Both of them agree well with what is predicted from Equation (6). Significant results previously reported in the literature are listed in Table 1.

In addition to all these nonmetallic MFBGs, metallic gratings have also been proposed for RI sensing. The existence of metal causes light to be coupled to modes of different properties [32,42]. A metal-dielectric-hybrid grating (Figure 2(b)) showed RI sensitive (a in Figure 7) and insensitive (d in Figure 7) behavior for different resonant modes (see inset of Figure 7) [32]. S_a of the sensitive channel (125 nm/RIU) is one order of magnitude larger than that of a nonmetallic MFBG with the same radius whereas S_a of the insensitive channel (8 nm/RIU) is one order of magnitude smaller. This can be attributed to the fact that the introduction of metal film causes the MF to support both surface guided modes (which have a larger modal overlap with the ambient medium, Figure 7(a,b)) and bound modes (where most of the energy is located in the dielectric core, Figure 7(c,d)).

Although thermal post-processing and hydrogen loading have been shown to induce grating capable of standing temperatures as high as 1,300 °C in conventional fibers [52]; for uFMBGs, only small temperature ranges have been detected because the photosensitized index modulation is unstable at high temperatures. In MF thermometers, up to now, only sMFBGs have been reported operating above 200 °C [31,35]. The sensitivity of these components is around 20 pm/°C, similar to the value predictable using Equation (7). Figure 8 is the experimental characterization of the sFMBG demonstrated using the sample shown in Figure 2(a). As the temperature increases, the Bragg wavelength red shifts. The extremely short length of the MFBG (∼36.6 μm) and wide operating range (∼20–450 °C) presents it a promising candidate for detecting temperature change in ultra-small spaces.

Although S_S remains almost the same for different MF diameters [26], S_F varies with the MF radius according to Equation (10). A MFBG with diameter of 3.5 μm reaches a force sensitivity of ∼1900 nm/N, which is more than three orders of magnitude compared to that of a conventional fiber [26]. If the Bragg wavelength can be detected to an accuracy of 0.05 nm, it would be possible to measure forces in the order of 10⁻⁵ N. For the sample reported in Figure 2(d), where the silica constitute only a small fraction of the MFBG cross section, a further three orders of magnitude improvement in sensitivity is predicted, with S_F reaching values in excess of 10⁶ nm/N, corresponding to forces of the order of nN. The MFBG strain/force sensors could offer attractive properties monitoring strain/force changes in power plant pipelines, airplane wings, and other civil engineering structures.