  We propose deep-subwavelength optical waveguides based on metal-dielectric
multilayer indefinite metamaterials with ultrahigh effective refractive
indices. Waveguide modes with different mode orders are systematically analyzed
with numerical simulations based on both metal-dielectric multilayer structures
and the effective medium approach. The dependences of waveguide mode indices,
propagation lengths and mode areas on different mode orders, free space
wavelengths and sizes of waveguide cross sections are studied. Furthermore,
waveguide modes are also illustrated with iso-frequency contours in the wave
vector space in order to investigate the mechanism of waveguide mode cutoff for
high order modes. The deep-subwavelength optical waveguide with a size smaller
than {\lambda}0/50 and a mode area in the order of 10-4 {\lambda}02 is
realized, and an ultrahigh effective refractive index up to 62.0 is achieved at
the telecommunication wavelength. This new type of metamaterial optical
waveguide opens up opportunities for various applications in enhanced
light-matter interactions.
