## 1 Introduction  
Optical surface waves (SWs), which are a special class of electromagnetic waves propagating along the interface of two dissimilar materials with fields decaying exponentially away from the interface, have been extensively studied in the last century. Owing to their characteristics of field enhancement [1] and localization [2], and high sensitivity in partner materials [3], SWs have shown promising potential in many applications, such as optical sensing [4], integrated optics [5], nano-optical tweezers [6], microscopy [7] and communication [8]. However, the introduction of these SWs is difficult since the fields of SWs should be evanescent in both partner media with Maxwell's equations satisfied at the boundary simultaneously. Typically, SWs can be realized when the partner media have opposite signs of dielectric functions [1], which leads to surface plasmon polaritons (SPPs) [9], or at least one of the partner media is anisotropic, which leads to Dyakonov SWs [10; 11], or one of the partner media is periodically inhomogeneous in the direction normal to the interface, which leads to Tamm SWs [12; 13]. With the prosperity of nanotechnology in the last two decades, manufactured metamaterials, including shallow metallic gratings [14], pillar structures [15] and multilayers [16], provide more possibilities for new types of SWs, such as Dyakonov plasmons [17], which take over traits from both Dyakonov SWs and SPPs [18], and Dyakonov-Tamm waves [19; 20], which combine the aspects of Dyakonov and Tamm SWs. It should be noted that most of these types of SWs are forward waves, in which the vector of energy flow and the corresponding wave share the same direction. Reports on backward SWs, of which the direction of energy flow is opposite to the wave vector, are relatively limited. However, backward SWs possess particular properties, such as high potential in surface-plasmon-based guiding structures [21], and can be directly demonstrated by the electromagnetic field detection around the photonic crystal surface with a local probe [22]. To fully take advantages of backward SWs, further investigation is required. The multilayered metal-dielectric metamaterials (MMDMs), which are artificial one-dimensional composites of periodically alternating dielectric and metallic layers, can be engineered so as to constitute an ideal platform for the implementation of unconventional surface states. SWs guided by a homogeneous dielectric medium and an MMDM [23; 24] can be regarded as a generalization of Tamm state, when the metallic and dielectric layers of the MMDM are parallel to the interface separating the two materials [25]. Electromagnetic analogue of Tamm state localized at the interface of different partner media has been extensively researched, including two dissimilar photonic crystals [26], an MMDM in contact with a left-handed metamaterial [27], two dissimilar MMDMs [28], or a hyperbolic metamaterial heterostructure consisting of stacked graphene sheets separated by dielectric layers [29]. Under the condition of long-wavelength approximation [30; 31], MMDMs are conventionally homogenized as uniaxial media through the effective media approximation (EMA) [32; 33; 34], which simplifies the procedure of analyzing the electromagnetic response of the MMDMs and provides an intuitive physical insight. Nevertheless, the description of MMDMs through the EMA method is incomplete, as the actual periodicity of MMDMs is not taken into account under the homogenization. It has been reported that both the thickness [35] and the order [24] of the metallic and dielectric layers have significant influence on the optical nonlocality of MMDMs due to the excitation of SPPs, thus resulting in the dispersions and diffraction properties of SWs supported by MMDMs that may severely deviate from the results acquired by the EMA method. In particular, additional backward SWs beyond the prediction of EMA method arise [28].