The emergence of metamaterials with artificially engineered subwavelength composites offers a new perspective on light manipulation and exhibits intriguing optical phenomena such as negative refraction [1][2][3], sub-diffraction imaging [4][5][6], invisible cloaking [7][8][9] and high index of refraction [10][11][12]. One unique kind of optical metamaterials is the indefinite metamaterial with extreme anisotropy, in which not all the principal components of the permittivity tensor have the same sign [13].The non-magnetic design and the off-resonance operation of the indefinite metamaterial can considerably reduce the optical absorption associated with conventional metamaterials [14]. Since optical waveguides play an important role in many fundamental studies of optical physics at nanoscale and in exciting applications in nanophotonics, optical waveguides based on indefinite metamaterials have been recently studied [19,20, 23,24,[26][27][28][29], in order to obtain novel optical properties beyond the conventional dielectric waveguides, especially slow light propagation [19,20], surface modes guidance [28,29], as well as subwavelength mode compression [23,24]. In this paper, we propose deep-subwavelength optical waveguides based on metal-dielectric multilayer indefinite metamaterials, which support waveguide modes with tight (1 )where f m is the volume filling ratio of silver, ε d and ε m are the permittivity corresponding to germanium and silver, respectively. The permittivity of germanium is ε d = 16, and the optical properties of silver are described by the Drude model Since large wave vectors are supported in indefinite metamaterials due to the hyperbolic dispersion, ultrahigh refractive indices can be reached, which will enable the formation of optical waveguides with deep subwavelength cross sections based on the total internal reflection (TIR) at the interface between metamaterial and air, as illustrated in Fig. 1(a). gives the distributions of optical field components for (1, m y ) mode calculated with the effective medium method, which agree very well with the multilayer results in Fig. 2(a). where k 0 is the free space wave vector corresponding to the free space wavelength λ 0 .The components of effective refractive index n eff are related to wave vector components along different directions, as (n eff,x , n eff,y , n eff,z ) = (k x /k 0 , k y /k 0 , k z /k 0 ). The 3D hyperboloid iso-frequency contour (IFC) of indefinite metamaterial in k-space for a specific λ 0 is shown in Fig. 4(a Fig. 4 (b) to the y direction in Fig. 4(c). According to Fig. 4(b), there are no mode cutoff for the (1, m y ) modes. However, the (3, 1) and (3, 2) modes have cutoff at λ 0 = 1.55 μm in Fig. 4(c), since k z is not available for the given mode orders.For λ 0 = 1 μm, the dispersion curve has much lower k y , so that all the (3, m y ) modes still exist.The effects of waveguide cross sections on the mode cutoff are then studied at a fixed wavelength of λ 0 = 1.55 μm. By varying the waveguide height L y or the 8 waveguide widt...