  Optical fiber with high numerical aperture (NA) can efficiently relieve the
degeneracy of higher order linearly polarized modes. The degeneracy relief is
investigated in two types of high NA fibers, i.e., low index cladding fiber and
high-index-core fiber. A naked core fiber, as with low index cladding, can be
used theoretically to generate the orbital angular momentum mode (OAMM) HE21
and the cylindrically symmetric modes (CSMs) TM01 and TE01. A high index core
fiber incorporated with high contrast index structure can be used similarly to
obtain OAMM HE31. The generation of both CSMs and OAMMs required tilted optical
gratings to couple the fundamental core mode HE11 into these modes. The tilt
angle and modulation period of the grating fringes can be calculated simply and
visually with the method proposed in this article.
