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Fig. 4. Overview of WGM resonators used for induced Fano resonance coupled into optical fibers.
with an optical fiber, considered to have a low Q-factor and to propagate a continuum mode, the capacity to induce Fano resonance is enhanced due to its high Q-factor, which represents the discrete mode [ 63 ]. Two mathematical methodologies have been developed for modeling the modal interference between both. The classical transfer matrix method analyzes the exchange of optical power between the resonator and the waveguide [ 64 ]. The coupled mode theory, proposed by Pierce at Bell Laboratories, accounts for different fields that perturb the system to calculate the spectrum [ 65 ].
In literature, a wide range of structures have been developed to propose a WGM. These include microspheres [ 66 ], microrings [ 67 ], microbubble [ 68 ], microdisks [ 69 ], microcapillaries [ 70 ], microcylinder [ 71 ], microtoroid [ 72 ] and others. The coupled waveguide exhibits a low Q-factor and operates within a limited frequency range, thereby inducing the Fano phenomenon. The structure can be obtained through various methods, which can be classified into two distinct groups. Single-cavity coupling, defined as a coupling configuration in which only one cavity is involved, can be achieved through unilateral coupling, a process in which a resonator is coupled to the side of a single waveguide. For example, a WGM microcavity coupled to a tapered fiber [ 73 ] exhibits the advantages of simple fabrication, high coupling efficiency, and elevated Q-value in the order of 10 6 [ 74 ] and a maximum obtained of 10 8 [ 75 ]. However, it is considered a fragile system. Alternatively, bilateral coupling typically involves positioning the resonator between two waveguides [ 76 ]. Another approach is the in-fiber coupling structure, which enhances system robustness by embedding the resonator within the fiber waveguide [ 77, 78 ]. However, the Q-factor presents lower values compared to microresonators coupled to tapered fibers, being limited to 10 4 [ 33 ]. This demands further investigations of this method in order to fully exploit the potential of WGM resonators and the robustness of optical fibers. An overview of the resonator’ s structures mentioned is illustrated in Figure 4.
Another common approach is multi-cavity coupling, which comprises the same coupling types showed above: unilateral, bilateral, and in-fiber; however, it involves more than one microcavity interacting through resonance [ 79 – 83 ]. In comparison, multi-cavity coupling more readily excites resonances between a low-Q mode and a high-Q mode due to the resonant interactions among the microcavities coupled to the fiber. Figure 5 presents typical configurations of these resonators with an optical fiber, which can be unilateral, bilateral, or in-fiber.
4.2 Microstructured optical fibers
Microstructured optical fibers are structures that differ from the classic single-mode fiber, in which different structures are developed within the fiber to guide light along the direction of propagation. Typically, instead of a central core with a higher refractive index, air holes are introduced within the fiber. Geometry and spatial organization are determinant for the confined and propagated modes [ 84 ]. The Fano resonance is induced when the mode confinement geometry is designed to enable interaction between a high- Q mode and a low-Q mode.
An investigation was conducted on how the phase relationship between each hollow-core cylinder in a photonic bandgap fiber induces Fano-like resonances in the scattering spectrum [ 15 ]. In that study, the number of cylinders and their spacing were identified as critical parameters in coupling terms and Fano appearance. Another confinement mechanism is the Inhibited Coupling( IC), which provide a wider transmission bandwidth than PBG [ 85 ]. Distinct MOFs were designed to achieve IC, and this mechanism has been used to obtain Fano resonance in Kagone fibers [ 86 ] and polygonal tube fibers [ 17, 19 ].
4.3
Interferometric strutures
Interferometric structures have been widely used in photonics for a range of applications. In the context of Fano resonance, this phenomenon has been observed in a Mach- Zender interferometer( MZI), and a theorical model was developed by Miroshinichenko et al. [ 87 ]. Several applications have been demonstrated using such configurations, including sensing with a tapered optical fiber [ 88 ], an in-line MZI incorporating an FBG inscribed in a D-shaped fiber [ 34 ], and microrings exhibiting a discrete comb-like spectrum coupled to arm of an MZI [ 55 ]. MZI-based Fano resonances have also been explored for optical bistability [ 89 ], and for tunable control of the Fano asymmetry using FBGs [ 44, 50, 54, 90 ], demonstrating a versatile approach to inducing Fano resonance in all-fiber systems.
Michelson interferometer configurations have been demonstrated in an in-fiber integrated setup with a microsphere WGM for sensing and lasing applications [ 33 ], as well as in a ring-cavity-coupled scheme for optical bistabil-