  We analyze the diffraction characteristics of dielectric gratings that
feature a high index grating layer, and devise, through rigorous numerical
calculations, large bandwidth, highly efficient, high dispersion dielectric
gratings in reflection, transmission, and immersed transmission geometry. A
dielectric TIR grating is suggested, whose -1dB spectral bandwidth is doubled
as compared to its all-glass equivalent. The short wavelength diffraction
efficiency is additionally improved by allowing for slanted lamella. The
grating surpasses a blazed gold grating over the full octave. An immersed
transmission grating is devised, whose -1dB bandwidth is tripled as compared to
its all-glass equivalent, and that surpasses an equivalent classical
transmission grating over nearly the full octave. A transmission grating in the
classical scattering geometry is suggested, that features a buried high index
layer. This grating provides effectively 100% diffraction efficiency at its
design wavelegth, and surpasses an equivalent fused silica grating over the
full octave.
