Long-period gratings (LPGs) in an optical fiber, which was designed and fabricated for achieving high insensitivity of the inscribed LPGs both to bending and to the external index of refraction, are investigated. The fiber has a double cladding, consisting of a uniform outer cladding and a graded-index inner cladding, where the index of refraction decreases approximately parabolically with radius. This fiber is referred to as the parabolic-index cladding (PIC) fiber. Results obtained in the preparation of preforms and drawing PIC fibers are given. The theoretical analysis, carried out by the mode expansion and beam propagation method, yields field distributions and effective indexes of the fundamental core mode LP01 and several low-order circularly symmetric cladding modes LP0m for several external indexes of refraction. Experiments are done with an LPG with a period of 214 μm inscribed with a CO2 laser in the PIC fiber. In experiments, transmission spectra of the LPG immersed in three liquids with refractive indexes in a relatively wide range around the index of silica (water, silica-index-matching liquid, silicone oil) are measured in a temperature range of 25-67.5°C. From experiments it follows, in agreement with the theoretical analysis, that the spectral positions of the investigated attenuation band at &lgr; ≈ 1550 nm (coupling between the LP01 and LP02 mode), determined at temperatures in the above range, are highly insensitive to the external index of refraction.