We designed, fabricated, and characterized a dielectric metamaterial lens created by varying the density of subwavelength low refractive index nanoholes in a high refractive index substrate, resulting in a locally variable effective refraction index. It is shown that a constructed graded index lens can overcome diffraction effects even when the aperture/wavelength (D/λ) ratio is smaller than 40. In addition to the conventional design of a polarization insensitive lens, we also show that a polarization diversity lens (f(o)≠f(e)) can be realized by arranging nanoholes in patterns with variable density in different transverse directions. Such a anisotropic microlens demonstrates polarization dependent focal lengths of 32 and 22 μm for linearly x- and y-polarized light, respectively, operating at a wavelength of λ=1550 nm. We also show numerically and demonstrate experimentally achromatic performance of the devices operating in the wavelength range of 1500-1900 nm with full width at half-maximum (FWHM) of the focal spots of about 4 μm.