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Fig. 7. Simulation of modal coupling efficiency:( a) influence of Fresnel reflectivity and( b) influence of the FBG reflected peak intensity on the resulting Fano profile.
Fresnel reflection at the fiber tip determined by the last term. The Fano parameter is controlled by the ratio between the length of the last pitch, U, and the Bragg period, K, of the FBG. For U K / 2, the asymmetric line shapes correspond to Fano parameters between �1 and 0. When U = K / 2, the Fano parameter equals 0, whereas for U K / 2, theFanoparameterliesbetween0and1.
P E r ¼ E r N K ðt K e ibK Þ 2k þ R U t 2Nþ2 ib 2Uþ2NK K e ð Þ; ð2Þ k¼0
" # E r ¼ E r Kð1 � ðt K e ibK Þ 2Nþ2 Þ þ R ð1 � ðt K e ibK Þ 2 U t 2Nþ2 K e ibð2Uþ2NKÞ: ð3Þ
Þ
Based on the analytical model, the coupling efficiency of the Fano-FBG is investigated as a function of both the grating reflectivity and the Fresnel reflection coefficient at the fiber tip. In this analysis, the grating length is kept constant at 1mm. InFigure 7a, the parameter R U( 0 – 3 %) quantifies the strength of the secondary reflection that couples with the grating-reflected field. As R U increases, the response evolves from a conventional, symmetric FBG spectrum( 0 %) to a clearly asymmetric Fano line shape( 3 %), evidencing progressively stronger interference. Figure 7b considers the Fresnel reflection associated with the silica – air interface at the fiber end, while the FBG reflectivity is swept from 0.1 % to 13 %. Increasing grating reflectivity reduces the relative contribution of the tip-reflected field, weakening the interference term; consequently, the grating-reflected component dominates and the Fano features progressively vanish. Conversely, higher tip reflectivity enhances the counter-propagating field, increasing the interference contrast and sharpening the Fano spectral asymmetry. Moreover, large grating reflectivity compresses the spectral bandwidth and drives the response toward the characteristic FBG peak, effectively masking the Fano profile. Therefore, achieving efficient modal coupling and inducing the Fano resonance requires a suitable trade-off between grating and tip reflectivities, ensuring comparable amplitudes of the two interfering reflected signals. Another relevant aspect is the tunability of the Fano asymmetric q- parameter via the phase shift introduced in the FBG. By adjusting the length of the final grating period, one can realize different interference regimes, yielding q = ± 1 and q = 0. Notably, thecaseq = 0 corresponds to a symmetric FBG notch response or anti-reflection AR-FBG, as reported in [ 37 ]. For q = ± 1, a representative example is provided by the 2024 study on the slice FBG approach [ 53 ], where this phase-shift produces a clearly asymmetric Fano profile.
4.4.2
Fano-like FBG applied as a sensing tool
In addition to the simulation work performed for the optical device, an experimental study was also carried out to evaluate its performance as a sensor for monitoring refractiveindex variations in liquids. Different water – sugar solutions were prepared to obtain refractive indices ranging from 1.334 to 1.339. The experiments were performed using a Fano-like FBG with q = �1( seeFig. 8a). The interrogation system was intensity-based, and self-referenced by measuring the intensity difference between the peak and the valley of the Fano profile; the experimental and numerical results are illustrated in Figure 8b. This approach was chosen because the Fresnel reflectivity strongly influences the signal intensity, while the peak wavelength exhibits low sensitivity to refractive-index variations. The experimental setup comprised broadband source with a spectral bandwidth of 100 nm centered at 1550 nm, a three-port optical fiber circulator( Thorlabs 6015-3) to route the reflected