A Review of Multimode Interference in Tapered Optical Fibers and Related Applications <s> A Singlemode-Tapered-Multimode-Singlemode (STMS) Structure and Its Fabrication <s> We report a reproducible technique of making side-polished fibers by embedding fibers in silicon V grooves and by polishing them mechanically. Details of V grooves and polishing techniques are described. The attenuation characteristics of polished fibers were measured by a liquid-drop method; the results are in excellent agreement with existing theoretical predictions. To facilitate comparisons, we cast expressions for the attenuation constant in terms of three generalized parameters: the V and b parameters for the fiber and a new generalized parameter V(ex) for the external medium. By using these generalized parameters, we can study the effects of the external medium on the attenuation constant of side-polished fibers in great detail, including in particular the region where the attenuation changes precipitously. <s> BIB001 </s> A Review of Multimode Interference in Tapered Optical Fibers and Related Applications <s> A Singlemode-Tapered-Multimode-Singlemode (STMS) Structure and Its Fabrication <s> A new beam-propagation method is presented whereby the exact scalar Helmholtz propagation operator is replaced by any one of a sequence of higher-order (n, n) Pade approximant operators. The resulting differential equation may then be discretized to obtain (in two dimensions) a matrix equation of bandwidth 2n + 1 that is solvable by using Standard implicit solution techniques. The final algorithm allows (for n = 2) accurate propagation at angles of greater than 55 deg from the propagation axis as well as propagation through materials with widely differing indices of refraction. <s> BIB002 </s> A Review of Multimode Interference in Tapered Optical Fibers and Related Applications <s> A Singlemode-Tapered-Multimode-Singlemode (STMS) Structure and Its Fabrication <s> A new beam-propagation method is presented whereby the Pade approximant wide-angle propagation operator is factored into a series of simpler Pade (1, 1) operators, thus leading naturally to a multistep method whose component steps are each solvable by using readily available paraxiallike solution techniques. The resulting method allows accurate approximations to true Helmholtz propagation while incurring only a modest numerical penalty. In addition, the tridiagonal form of the component steps allows the straightforward use of the previously reported transparent boundary condition. <s> BIB003 </s> A Review of Multimode Interference in Tapered Optical Fibers and Related Applications <s> A Singlemode-Tapered-Multimode-Singlemode (STMS) Structure and Its Fabrication <s> We review the recent developments in the area of optical fiber grating sensors, including quasi-distributed strain sensing using Bragg gratings, systems based on chirped gratings, intragrating sensing concepts, long period-based grating sensors, fiber grating laser-based systems, and interferometric sensor systems based on grating reflectors. <s> BIB004 </s> A Review of Multimode Interference in Tapered Optical Fibers and Related Applications <s> A Singlemode-Tapered-Multimode-Singlemode (STMS) Structure and Its Fabrication <s> Recent research on fibre optic long-period gratings (LPGs) is reviewed with emphasis placed upon the characteristics of LPGs that make them attractive for applications in sensing strain, temperature, bend radius and external index of refraction. The prospect of the development of multi-parameter sensors, capable of simultaneously monitoring a number of these measurands will be discussed. <s> BIB005 </s> A Review of Multimode Interference in Tapered Optical Fibers and Related Applications <s> A Singlemode-Tapered-Multimode-Singlemode (STMS) Structure and Its Fabrication <s> A tunable photonic crystal fiber (PCF) coupler, which couples part of the optical power in one PCF with that in another PCF, has been made by side polishing. We fabricated the PCF coupler by mating two side-polished PCFs. We achieved evanescent field coupling between the core modes of the two PCFs by using side polishing to bring the cores close to each other. By adjusting the mating angle between the two side-polished PCFs we obtained as much as 90% tunability in the coupling ratio. The spectrum of the coupling ratio was almost flat, with small ripples, over a 400-nm wavelength range. <s> BIB006 </s> A Review of Multimode Interference in Tapered Optical Fibers and Related Applications <s> A Singlemode-Tapered-Multimode-Singlemode (STMS) Structure and Its Fabrication <s> Abstract When a cleaved optical fibre is immersed in hydrofluoric acid, the compositional variation within the core and the cladding results in the core-region being etched at a faster rate. This paper presents details of the experimental procedures that were used to etch SM 800 optical fibres and the characterization of the micro-cavities using a 3D surface profiler, optical and scanning electron microscopy. Two such acid-etched fibres were fusion spliced to create an intrinsic Fabry–Perot cavity. These strain sensors were surface-mounted on to fibre reinforced composite test specimens and subjected to quasi-static and cyclic loading. The intrinsic fibre Fabry–Perot sensors were interrogated using low-coherence interferometry. The output from the intrinsic Fabry–Perot sensors showed excellent correlation with surface-mounted electrical resistance strain gauges during quasi-static and cyclic loading. <s> BIB007 </s> A Review of Multimode Interference in Tapered Optical Fibers and Related Applications <s> A Singlemode-Tapered-Multimode-Singlemode (STMS) Structure and Its Fabrication <s> Mach-Zehnder and Michelson interferometers using core-offset attenuators were demonstrated. As the relative offset direction of the two attenuators in the Mach-Zehnder interferometer can significantly affect the extinction ratio of the interference pattern, single core-offset attenuator-based sensors appear more robust and repeatable. A novel fiber Michelson interferometer refractive index (RI) sensor was subsequently realized by a single core-offset attenuator and a layer of ~ 500-nm gold coating. The device had a minimum insertion loss of 0.01 dB and maximum extinction ratio over 9 dB. The sensitivity (0.333 nm) of the new sensor to its surrounding RI change (0.01) was found to be comparable to that (0.252 nm) of an identical long period gratings pair Mach-Zehnder interferometric sensor, and its ease of fabrication makes it a low-cost alternative to existing sensing applications. <s> BIB008 </s> A Review of Multimode Interference in Tapered Optical Fibers and Related Applications <s> A Singlemode-Tapered-Multimode-Singlemode (STMS) Structure and Its Fabrication <s> We designed and fabricated a multi-core fiber whose attenuation is less than 0.18 dB/km and effective area is about 80 µm2. Crosstalk was observed be less than −55.5 dB after 17.6 km propagation. <s> BIB009 </s> A Review of Multimode Interference in Tapered Optical Fibers and Related Applications <s> A Singlemode-Tapered-Multimode-Singlemode (STMS) Structure and Its Fabrication <s> All-fiber in-line single-mode—multimode—single-mode (SMS) and single-mode—tapered-multimode—single-mode (STMS) fiber structures are investigated. A wide-angle beam propagation method in cylindrical coordinates is developed and employed for numerical simulations of the light propagation performance of such fiber devices. The effect of strong mode interference on the performance of the devices is studied and verified numerically; results indicate that the proposed STMS structure can be exploited for measuring a broad refractive index range with reasonable high resolution, compared with the conventional SMS structure. <s> BIB010
A traditional SMS fiber structure cannot be considered as a suitable candidate for many sensing applications including RI and humidity, since the thick cladding of the MMF section limits the interaction between the light propagating in the MMF and the external environment. Therefore, in order to allow an SMS fiber structure to be utilized for detecting physical quantities such as RI, humidity, gas concentration and magnetic field strength, many post-process techniques have been proposed, which include hydrofluoric acid etching, side polishing, misalignment, bend, and grating writing BIB004 BIB007 BIB001 BIB008 BIB009 . In BIB010 we were first to propose an STMS fiber structure based on a fiber tapering technique. In this sub-section, a theoretical model and an introduction to the fabrication method of the STMS fiber structure are reviewed. The schematic diagram of the STMS fiber structure is depicted in the Figure 4a . It consists of a section of tapered silica MMF sandwiched between two standard SMFs. Figure 4b shows the corresponding illustration of an STMS fiber structure with relevant geometrical parameters where the uniform tapered waist ( ) and transitional length of the tapered MMF section are respectively expressed as: = ln( ) where is the initial, radius of the untapered MMF, and are respectively the radius and length of the uniform tapered waist region and is approximated to the length of the heating region .  A traditional SMS fiber structure cannot be considered as a suitable candidate for many sensing applications including RI and humidity, since the thick cladding of the MMF section limits the interaction between the light propagating in the MMF and the external environment. Therefore, in order to allow an SMS fiber structure to be utilized for detecting physical quantities such as RI, humidity, gas concentration and magnetic field strength, many post-process techniques have been proposed, which include hydrofluoric acid etching, side polishing, misalignment, bend, and grating writing BIB004 BIB007 BIB001 BIB008 BIB009 . In BIB010 we were first to propose an STMS fiber structure based on a fiber tapering technique. In this sub-section, a theoretical model and an introduction to the fabrication method of the STMS fiber structure are reviewed. The schematic diagram of the STMS fiber structure is depicted in the Figure 4a . It consists of a section of tapered silica MMF sandwiched between two standard SMFs. Figure 4b shows the corresponding illustration of an STMS fiber structure with relevant geometrical parameters where the uniform tapered waist R(Z) and transitional length Z 0 of the tapered MMF section are respectively expressed as:  where R 0 is the initial, radius of the untapered MMF, R w and L w are respectively the radius and length of the uniform tapered waist region and L w is approximated to the length of the heating region L 0 . corresponding illustration of an STMS fiber structure with relevant geometrical parameters where the uniform tapered waist ( ) and transitional length of the tapered MMF section are respectively expressed as: where is the initial, radius of the untapered MMF, and are respectively the radius and length of the uniform tapered waist region and is approximated to the length of the heating region . (a) (b) The light propagation characteristic within the STMS fiber structure can be simulated by employing a wide-angle beam propagation method (WA-BPM)  . Starting with a cylindrical WA-BPM in an SMS fiber structure and given the symmetrical nature of the SMS fiber structure, the scalar wave equation can be easily calculated by adopting a cylindrical coordinate system, thus: where k and n(r, z) respectively represent wave number in free space and the RI distribution of the MMF. Assuming a slowly varying envelop approximation for the BPM, such as E(r, z) =Ê(r, z) exp(jkn 0 z), where n 0 is the reference RI, then forÊ(r, z), the corresponding beam propagation equation can be calculated as follows: ∂Ê ∂z BIB006 where PÊ = Ê . With the Padé approximation BIB002 , a Crank-Nicolson finite-difference (FD) scheme and multistep method as developed in BIB003  , Equation (9) can be rewritten as:Ê l+1 = (1 + a n P)(1 + a n−1 P) where a i and a * i (i = 1, 2, · · · , n) are determined by the polynomials based on the Padé approximation. Therefore, for the i − th step,Ê l+ i n can be solved by: where . At the fiber axis, r = 0, the L'Hôpital rule can be applied and we have: Note that it is sufficient to calculate this model in half of the waveguide due to the circular symmetry of this fiber structure, which can save significant computational time and memory. The PML is also employed as the boundary condition of the computational region (r = R). Finally, the coupling loss of the light field E(r, z) at a propagation distance z can be expressed as: where f (r) denotes the fundamental mode of the input SMF. The output SMF is assumed to have the same fiber parameters as the input SMF. According to the equations above, the field distribution evolution along the tapered MMF section and coupling loss to the output SMF of the STMS fiber structure can be obtained, as shown in Figure 5a ,b. When the parameters of the tapered multimode fiber are substituted into the equations above, we obtain the optical amplitude distributions of the propagation fields within the STMS fiber structure and the relationship between the coupling loss to the output SMF and tapered MMF length. Comparing the simulated results of the SMS fiber structure and the STMS fiber structure, it is clear that for the STMS structures there are strong mode interferences in the tapered multimode fiber. The main reason for this is the focusing effects of the left taper transitional section. In addition, due to the tapering induced high order mode filtering effect, the mode numbers and interference intensity within the right taper transitional section are smaller than those of the SMS fiber structure. As regards fabrication of the STMS fiber structure, one approach is to utilise a CO2 laser irradiation technique BIB005 , as shown in Figure 6 . A CO2 laser (Model 48-2KWL Synrad, Mukilteo, WA, USA) and a ZnSe cylindrical lens with focal length of 254 ± 0.5% mm are employed in the fabrication process. It is noteworthy that the used CO2 laser has a maximum output power of 30 W at its working wavelength (circa 10.6 µm), and the ZnSe cylindrical lens used has focal length of 254 ± 0.5% mm. When the parameters of the tapered multimode fiber are substituted into the equations above, we obtain the optical amplitude distributions of the propagation fields within the STMS fiber structure and the relationship between the coupling loss to the output SMF and tapered MMF length. Comparing the simulated results of the SMS fiber structure and the STMS fiber structure, it is clear that for the STMS structures there are strong mode interferences in the tapered multimode fiber. The main reason for this is the focusing effects of the left taper transitional section. In addition, due to the tapering induced high order mode filtering effect, the mode numbers and interference intensity within the right taper transitional section are smaller than those of the SMS fiber structure. As regards fabrication of the STMS fiber structure, one approach is to utilise a CO 2 laser irradiation technique BIB005 , as shown in Figure 6 . A CO 2 laser (Model 48-2KWL Synrad, Mukilteo, WA, USA) and a ZnSe cylindrical lens with focal length of 254 ± 0.5% mm are employed in the fabrication process. It is noteworthy that the used CO 2 laser has a maximum output power of 30 W at its working wavelength (circa 10.6 µm), and the ZnSe cylindrical lens used has focal length of 254 ± 0.5% mm. within the right taper transitional section are smaller than those of the SMS fiber structure. As regards fabrication of the STMS fiber structure, one approach is to utilise a CO2 laser irradiation technique BIB005 , as shown in Figure 6 . A CO2 laser (Model 48-2KWL Synrad, Mukilteo, WA, USA) and a ZnSe cylindrical lens with focal length of 254 ± 0.5% mm are employed in the fabrication process. It is noteworthy that the used CO2 laser has a maximum output power of 30 W at its working wavelength (circa 10.6 µm), and the ZnSe cylindrical lens used has focal length of 254 ± 0.5% mm. The practical step involved firstly fabricating a traditional SMS fiber structure by splicing a specific length of uncoated MMF (ASF 105/125) with two standard SMFs with the help of a high precision cleaver and a commercial fusion splicer. Then this SMS fiber structure is fixed, taking care to avoid any bending, on two three-dimensional (3D) translation stages which can precisely move the whole fiber structure and thus adjust the heating position of this fiber structure. Meanwhile, two 3g-weights are hung on the SMF structure to ensure that the SMS fiber structure The practical step involved firstly fabricating a traditional SMS fiber structure by splicing a specific length of uncoated MMF (ASF 105/125) with two standard SMFs with the help of a high precision cleaver and a commercial fusion splicer. Then this SMS fiber structure is fixed, taking care to avoid any bending, on two three-dimensional (3D) translation stages which can precisely move the whole fiber structure and thus adjust the heating position of this fiber structure. Meanwhile, two 3g-weights are hung on the SMF structure to ensure that the SMS fiber structure remains straight with the help of a constant applied tension. In this experiment, the position of the laser beam can be adjusted flexibly by using gold-coated mirrors on a one-dimensional (1D) motorized translation stage, while the opening and closing of its shutter can be controlled through a LabView program. When the output power of CO 2 laser is fixed at 15 W, the CO 2 laser beam irradiation can produce a tapered MMF.