  Significant suppression of radiation in 3D structured media with small
refractive index 1.4-1.6, such as of glass or polymers, is a desirable feature
yet to be obtained. For periodical structures this is realised at frequencies
of the complete photonic band gap (CPBG), which up to now was demonstrated to
open for materials with refractive index of at least 1.9. We present here a
quasiperiodic 3D structure consisting of multiple overlapping gratings with a
homogeneous distribution of Bragg peaks on a sphere in reciprocal space, which
allows efficient suppression of emission. Recently we have presented the
theoretical model, considering interaction with the neighbouring gratings only,
that estimates a finite CPBG for arbitrarily small refractive indices and thus
complete emission suppression in infinite structures. However, numerical
simulations demonstrate a finite leakage of power from emitter not predicted by
the model. Still the simulations show -10 dB suppression in 3D structures with
optimised number of gratings. Astonishingly, as we show here, this limit is
almost independent of the refractive index contrast. Also, the structures with
a defined number of gratings show maximal suppression at certain refractive
indices, losing the suppression even at higher refractive indices. The -10 dB
suppression is demonstrated for refractive index contrast as low as 1.30.
