Analysis of volume-phase-grating characteristics that a variety of refractive index distributions influence. Part 1: derivation of coupling coefficients and grating thickness, and characteristics of the G-factors

Analysis of volume-phase-grating characteristics that a variety of refractive index distributions influence. Part 2: Analysis of spectral bandwidths and angular bandwidths, and the derivation of these equations

To expand the spectral bandwidths with high diffraction efficiency of volume phase (VP) gratings, it is important to know the influence of a variety of refractive index distributions (RIDs) inside the recording material on the bandwidths. The influence of various graded types of RIDs on the full width at half the maximum of the spectral bandwidth (${\lambda _{\rm FWHM}}$λ_FWHM) and the angular bandwidth (${\theta _{\rm FWHM}}$θ_FWHM) is investigated in the Bragg regime. It becomes clear that various graded types of RIDs influence their bandwidths and the characteristics, and the combination of the RIDs and refractive index modulation (${{ n}_m}$n_m) is the most important factor for obtaining the large bandwidths of VP gratings. Theoretically, the ${\lambda _{\rm FWHM}}$λ_FWHM are obtained as the values of 149.7 nm and 257.6 nm (${{ n}_m} = {0.045}$n_m=0.045) for the wavelengths of 900 nm and 1550 nm, respectively. Furthermore, I have succeeded in developing the expression for ${\lambda _{\rm FWHM}}$λ_FWHM and ${\theta _{\rm FWHM}}$θ_FWHM, using the proportional characteristics of VP gratings investigated in this work. The correlative coupling-length coefficients "G-factors" devised as a variety of RID functions in the previous paper are introduced into them. It is apparent that the derived equations of the bandwidths can be adapted to any graded type of RID that changes continuously from a triangular type to a rectangular type of VP grating, which has not been reported yet. The equations have made it easy to design ${\lambda _{\rm FWHM}}$λ_FWHM and ${\theta _{\rm FWHM}}$θ_FWHM of VP gratings.