Fig. 1. Schematic diagram of LPFG

Fig. 2. LPFG. (a) Three-layer structure model; (b) transverse refractive index

Fig. 3. Change of effective refractive index of core fundamental mode with wavelength

Fig. 4. Intensity distributions of the 8 lower order core modes

Fig. 5. Change of effective refractive index of cladding modes with wavelength for the first 19 times before the first diffraction order

Fig. 6. Intensity distributions of four low order cladding modes excited by circularly symmetric refractive index modulation of LPFG across the fiber cross-section

Fig. 7. Variation of coupling constants between fundamental core mode and cladding mode on the cladding mode order

Fig. 8. Transmission spectra and mode pattern of LPFG fabricated in B-Ge-codoped single-mode fiber

Fig. 9. Phase matching curves of the cladding modes of LPFG. (a) LP₀₂ mode to LP_0,20 mode; (b) LP_0,11 mode to LP_0,20 mode

Fig. 10. Schematic diagram of the LPFG inscription by the UV laser techniques. (a) Amplitude mask exposure technique; (b) point by point inscribing method^[19]

Fig. 11. Typical transmission spectrum of a carbon dioxide laser written LPFG in a commercial B-Ge-codoped fiber (together with an image of the near-field pattern of the coupled LP cladding mode at the resonance wavelength of 1531 nm)^[7]

Fig. 12. Schematic diagram of the experimental setup for the fabrication of helical LPFG with carbon dioxide laser^[30]

Fig. 13. Schematic diagram of experimental setup for LPFG fabrication technique with femtosecond laser^[41]

Fig. 14. Schematic diagram of arc-induced LPFG^[47]

Fig. 15. Side view of a mechanically induced LPFG^[52]

Fig. 16. Transmission spectra of mechanically induced LPFG with increasing applied pressure^[52]

Fig. 17. Schematic diagram of HF etched LPFG^[58]

Fig. 18. Evolution of transmission spectra when strain is applied to the etched LPFG^[58]

Fig. 19. Schematic diagram of the experimental setup for fabrication of the micro-tapered LPFG^[66]

Fig. 20. Schematic diagram of experiment. (a) Schematic of alignment of metal mask and fiber; (b) cross section of optical fiber after He ion implantation^[71]

Fig. 21. Schematic diagram of the acoustic induced LPFG^[77]

Fig. 22. Dependence of the grating contrast on the laser scanning cycle of carbon dioxide laser when the fiber is applied with external strain during the fabrication of the LPFG^[26]

Fig. 23. Schematic diagram of the relation between glass volume and heating temperature for the fiber being heated and cooled at different conditions^[7]

Fig. 24. Dual-dip resonance of LPFG at dispersion turning point

Fig. 25. Transmission spectra of LPFG. (a) Single LPFG; (b) cascaded LPFGs ^[86]

Fig. 26. Temperature characteristics of conventional LPFG with LP₀₆ cladding mode^[86]

Fig. 27. Temperature characteristics of coated LPFG with LP₀₆ cladding mode. (a) Silicone rubber; (b) UV curing adhesive^[86]

Fig. 28. LPFG inscribed in photonic crystal fiber by carbon dioxide laser^[24]

Fig. 29. Strain characteristics of the carbon dioxide laser written LPFG with parallel inclined planes^[92]

Fig. 30. Strain characteristics of the fast and slow axis modes of the helical LPFG written in polarization-maintaining fiber^[33]

Fig. 31. Bending characteristics of cascaded tilted long-period fiber gratings^[97]

Fig. 32. Structure of concave-lens-like long-period fiber grating^[99]

Fig. 33. Bending characteristics of the LPFG written in the polarization-maintaining fiber. (a) Fast axis mode; (b) slow axis mode

Fig. 34. Twist characteristics of the LPFG inscribed in the double-cladding fiber by carbon dioxide laser^[101]

Fig. 35. Twist characteristics of two-mode fiber LPFG. (a) Conventional LPFG; (b) tilted LPFGs^[9]

Fig. 36. Twist characteristics of two-mode fiber LPFG. (a) Helical LPFG^[30]; (b) chiral LPFG^[32]

Fig. 37. Twist characteristics of polarization-maintaining fiber LPFG^[33]

Fig. 38. SRI characteristics of LPFG near dispersion turning point^[6]

Fig. 39. Relationship between fiber cladding diameter and the period of LPFG^[114]

Fig. 40. SRI characteristics of the LPFG written in the thin-cladding fiber^[114]

Fig. 41. SRI characteristics of LPFG before and after the etching^[115]

Fig. 42. The pH sensing characteristics of the PAH/PAA-coated LPFGs with and without clad etching^[115]

Fig. 43. Schematic diagram of oxide film coated LPFG for environmental refractive index sensing^[118]

Fig. 44. SEM picture of the cross section of the LPFG coated with Al₂O₃ nano-film^[27]

Fig. 45. SRI characteristics of LPFG coating with Al₂O₃ nano film^[27]

Fig. 46. SRI characteristics of LPFG coating with TiO₂ nano film^[122]

Fig. 47. SEM images of the coated LPFG. (a) Fiber coated with pure PVA film; (b) fiber coated with PEG/PVA composite film; (c) cross-section of the coated LPFG; (d) film thickness of the coated film^[123]

Fig. 48. Dependence of the resonance wavelength under the ascending and descending humidity process of PEG/PVA coated LPFG^[123]

Fig. 49. Experimental setup of magnetic field sensor based on LPFG and magnetic fluid^[124]

Fig. 50. Transverse profile schematic diagram of thin cladding polarization maintaining fiber immersion into magnetic fluid^[125]

Fig. 51. Magnetic field characteristics of thin cladding polarization maintaining fiber immersion into magnetic fluid. (a) Fast axis mode; (b) slow axis mode^[125]

Fig. 52. Diagram of the cascaded LPFG interferometer and spectrum of the interferometer^[126]

Fig. 53. Schematic diagram of cascaded coated long period fiber grating interferometer^[83]

Fig. 54. Transmission spectra of cascaded polarization-maintaining fiber LPFG

Fig. 55. Sensing characteristics of cascaded polarization-maintaining fiber LPFGs. (a) Temperature; (b) strain

Fig. 56. Schematic diagram of EDFA gain spectrum based on LPFG gain equalizer^[129]

Fig. 57. LPFG. (a) Schematic of two and three phase-shifted LPFGs; (b) illustration of passband characteristics of filters based on LPFGs^[133]

Fig. 58. Tunable all-fiber band rejection filters. (a) Schematic of the bandwidth-tunable all-fiber band rejection filters based on a carbon dioxide laser-induced helicoidal LPFG pair of opposite helicities; (b) spectral characteristic of the fabricated bandwidth-tunable all-fiber band rejection filters according to the rotation angle ^[136]

Fig. 59. Tunable bandwidth all-fiber rejection filters. Schematic diagrams of (a1) LPFG; (a2) R-LPFG; (b) spectral characteristics^[138]

Fig. 60. Cascaded helical LPFGs written by carbon dioxide laser^[34]

Fig. 61. Top view of the V groove holder that was used to keep part of the helical LPFG simmered in oil^[141]

Fig. 62. Experimental diagram of fabrication of acousto-optic grating and generation of first order OAM ^[143]

Fig. 63. Far field patterns measured at the output of the acousto-optic mode converter. (a) LP₀₁ mode; (b) LP₁₁ mode; (c) LP₂₁ mode; (d) LP₀₂ mode^[145]

Fig. 64. Experimental setup. (a) Experimental equipment diagram of fabrication of micro-bending grating and generation of OAM; (b) transmission spectrum of LP₀₁-LP₁₁ mode converter by the grating; (c) pattern and interference pattern of 1st order OAM ^[146]

Fig. 65. LPFG. (a) Spectrum of the LPFG with 15-period gratings; (b) mode field distribution at different wavelengths^[149]

Fig. 66. Spot pattern and interference pattern at the output of (a) LPFG and (b) tilted LPFG^[9]

Fig. 67. Mode converter based on LPFG. (a) Spectrum of a single mode converter based on a LPFG; (b) mode field distribution and interference pattern of a single mode converter; (c) spectrum of the cascaded mode converter; (d) intensity distribution and interference patterns of the cascaded mode converter ^[150]

Fig. 68. OAM mode converter based on PCF. (a) Interference patterns of the OAM₊₆ mode generated by the helical PCF; (b) interference patterns of the OAM₊₅ mode generated by the helical PCF ^[36]

Fig. 69. Transmission spectra and intensity distributions of the PM-PCF ^[151]

Fig. 70. (a) Schematic diagram of the cascaded few-mode fiber LPFGs; (b) transmission spectra of the cascaded few-mode fiber LPFGs with different grating periods in 1.55 μm and 2 μm waveband^[154]

Fig. 71. LPFG-based polarizer. (a) PDL; (b) polarization extinction ratio^[160]

Fig. 72. Transmission spectra of a 48 mm 45° TFG measured using a single wavelength at 1550 nm at two orthogonal polarization states (P1 and P2) ^[163]

Fig. 73. PDL and transmission spectra of helical LPFG with a polarization-preserving fiber LPFG with a period of 480 μm ^[33]

Fig. 74. Principle of operation of the wavelength-selective coupler based on LPFG ^[165]

Fig. 75. Evanescent-field coupling between a LPFG and an identical bare fiber and mode distributions

Fig. 76. Design of broadband coupler based on three parallel identical LPFGs ^[169]

Fig. 77. Schematics of a voltage-controllable coupler based on two LPFGs with divided coil heaters^[171]

Fig. 78. Add-drop multiplexer using two mode couplers based on LPFGs ^[165]

Fig. 79. Wavelength-tunable add/drop multiplexer by using four identical LPGs and PZT fiber stretchers ^[172]

Fig. 80. Total conversion ratio of MADMs for LP₀₂ mode and LP₀₃ mode^[174]