Fig. 1. CV, LP and OAM modes in few-mode fibers

Fig. 2. Schematic diagram of FM-LPFG

Fig. 3. Experimental setup for fabricating FM-LPFG using CO₂ laser method^[48]

Fig. 4. Experimental setup for fabricating FM-LPFG using femtosecond laser method^[66]

Fig. 5. Experimental setup for fabricating FM-LPFG using mechanical micro-bending method^[52]

Fig. 6. Experimental setup for fabricating FM-LPFG using acoustically-induced method^[55]

Fig. 7. Experimental setup for fabricating FM-LPFG using hydrogen‒oxygen flame heating method^[64]. (a) Schematic diagram of the setup; (b) periodic helical structure; (c) scanning electron micrographs of the sides and cross sections

Fig. 8. Experimental setup for fabricating FM-LPFG using arc-discharge method^[62]

Fig. 9. Schematic of the principle and challenges of higher-order mode generation using FM-LPFG. (a) Schematic of FM-LPFG fabricated by CO₂ laser with one-side exposure; (b) refractive index modulation; (c) effective mode refractive index difference between the fundamental mode and the first- to fourth-order modes increases with the increase of the mode order; (d) relative coupling coefficients between the fundamental mode and the first- to fourth-order modes decreases with the increase of the mode order

Fig. 10. Generation of first-order mode by preparing FM-LPFG using various methods. (a) CO₂ laser method^[43,48]; (b) femtosecond laser method^[66]; (c) hydrogen‒oxygen flame heating method^[64]; (d) mechanical micro-bending method^[52]; (e) acoustically-induced method^[55]; (f) arc-discharge method^[62]

Fig. 11. Generation of higher-order modes in FM-LPFG using the strong modulated method^[30]. (a) Schematic diagram of the strong modulated FM-LPFG; (b) transmission spectrum resulting from fundamental mode conversion to third-order mode and mode distributions before and after conversion; (c) side view and cross-section of the strong modulated FM-LPFG; (d) mode intensity distributions and interference patterns of the OAM_±3 modes

Fig. 12. Generation of higher-order modes in FM-LPFG using the indirect conversion method^[35,84]. (a) Model schematic, transmission spectra resulting from fundamental mode indirect conversion to second-order mode, as well as mode distributions before and after conversion; (b) calculated coupling coefficient when the source modes are LP₀₁, LP₁₁, LP₂₁, and LP₃₁; (c) transmission spectra and mode distributions of second to fourth-order modes using the indirect conversion method

Fig. 13. Generation of higher-order modes in FM-LPFG using the helical grating or chiral grating method^[75]. (a) Schematic diagram of the experimental setup; (b) schematic diagram of the chiral grating; (c) transmission spectra resulting from fundamental mode conversion to first-, second-, and third-order modes; (d) mode distributions and interference patterns for the first- to third-order modes

Fig. 14. Generation of higher-order modes in FM-LPFG using the preset-twist method^[40,76-78]. (a) Schematic diagram of the preset-twist FM-LPFG; (b) effect of different twist angles on the refractive index modulation required for grating fabrication; (c) transmission spectra resulting from fundamental mode conversion to second-, third- , and fourth-order modes; (d) mode distributions and interference patterns of the second- to fourth-order OAM modes

Fig. 15. Broadband mode generation realized in FM-LPFG using dual-resonance coupling method^[92]. (a) Schematic diagram of the FM-LPFG combining the dual-resonance effect and fusion taper technique; (b) conversion efficiency with cladding diameters of 113 μm, 121 μm, and 125 μm; (c) first-order mode distributions at different wavelengths; (d) variance in the dispersion turning point with the taper ratio

Fig. 16. Broadband mode generation in FM-LPFG using the decreasing period number method^[96]. (a) Transmission spectra obtained by decreasing the period number from 30 to 8; (b) first-order mode distribution at different wavelengths after passing grating

Fig. 17. Broadband mode generation in FM-LPFG using chirped grating method^[97,100]. (a) Conventional chirped grating; (b) tilted chirped grating

Fig. 18. Broadband mode generation in FM-LPFG using phase-shift grating method^[101]. (a) Schematic structure of the phase-shift grating; (b) transmission spectra during the fabrication process; (c) intensity distributions and interference patterns of the OAM modes at different wavelengths

Fig. 19. Broadband mode generation in FM-LPFG using cascaded grating method^[77]. (a) Schematic of the cascaded grating structure; (b) transmission spectra of the broadband third-order mode converter; (c) intensity distributions and interference patterns of the third-order mode at different wavelengths

Fig. 20. Multi-channel mode generation in FM-LPFG using mode selective interferometer^[106]. (a) Schematic diagram of multi-channel MSI; (b) intensity and interference patterns of second-order modes at different wavelengths; (c) transmission spectra in twisted and twist-free states

Fig. 21. Multi-channel mode generation using parallel FM-LPFG^[80]. (a) Schematic diagram of the parallel FM-LPFG; (b) transmission spectra of parallel FM-LPFG under different parameters; (c) mode distributions of parallel FM-LPFG at the resonance wavelengths

Fig. 22. Multi-channel mode generation using high diffraction order FM-LPFG^[109]. (a) Schematic diagram of the high diffraction order FM-LPFG; (b) transmission spectrum of multi-channel high-order mode converter and mode distributions at resonance wavelengths

Fig. 23. Multi-channel mode generation using cascaded FM-LPFG^[40]. (a) Schematic diagram of the cascaded FM-LPFG; (b) transmission spectrum of multi-channel high-order mode converter and mode distributions at resonance wavelengths

Fig. 24. Schematic diagram of generation for structured light using FM-LPFG

Fig. 25. High-order continuous light output of fiber laser using FM-LPFG^[117]. (a) Schematic diagram of FM-LPFG for realizing vector mode coupling; (b) schematic diagram of the cylindrical vector beam fiber laser; (c) transmission spectra of FM-LPFG and intensity distributions of output beam; (d) input-output power characteristic of the two arms

Fig. 26. High-order pulsed light output of fiber laser using FM-LPFG^[125]. (a) Diagram of the mode-locked fiber laser; (b) output spectra before and after FM-LPFG at single soliton operation; (c) TE₀₁ and TM₀₁ mode distributions after FM-LPFG at single-soliton operation; (d) pulse traces comparison before and after FM-LPFG

Fig. 27. Temperature sensing measurements using FM-LPFG^[134]. (a) Schematic of the fabrication process of coating the FM-LPFG after fusion tapering; (b) offsets of the two resonant peaks with increasing temperature and the fitted curves between wavelength and temperature

Fig. 28. Twist sensing measurements using FM-LPFG^[71]. (a) Resonance wavelength shift versus twist rate for the SM-LPFG and SM-HLPG; (b) dependence of resonance wavelength shift on the twist rate for the FM-HLPG

Fig. 29. Strain sensing measurements using FM-LPFG^[133]. (a) Strain sensitivity of the FM-LPFG at non-dispersion turning point; (b) strain sensitivity of two dips at the dispersion turning point

Fig. 30. Simultaneous multi-parameter sensing measurement using FM-LPFG^[62]. (a) Experimental setup for temperature and strain measurement; (b) transmission spectra of FM-LPFG; (c) transmission spectra of two dips at different temperatures and the corresponding wavelength of two dips shifts as a function of temperature; (d) transmission spectra of two dips at different strains and the corresponding wavelength of two dips shifts as a function of strain