## 1 Introduction

Surface electromagnetic waves, propagating along the interface of two dissimilar media, have been the subject of extensive research during the last decades as they represent one of the fundamental concepts of nanophotonics. Understanding the optical properties of surface waves is of great importance for realizing their practical application.

There are several types of surface waves that differ in material type, domain of existence, propagation constant, decay profile. Among different types of surface waves, there are surface plasmon-polariton at a metal-dielectric interface [1], Tamm surface states at a photonic crystal boundary [2], [3], [4], surface solitons at a nonlinear interface [5], and many others.

Later, different combinations of isotropic, uniaxial, biaxial, and chiral materials have been demonstrated to support DSWs [8], [9], [10], [11], [12], [13], [14].

A narrow range of propagation angles makes the experimental observation of DSWs rather complicated [15]. As a result, the first detection of these waves has been demonstrated only in 2009 [16]. The authors used Otto-Kretschmann configuration to observe Dyakonov surface states at the interface of a biaxial crystal and an isotropic liquid. Another perspective approach to obtain Dyakonov-like surface waves experimentally is the usage of partnering thin films between anisotropic and isotropic media [17]. In such systems, the direction of hybrid Dyakonov-guided modes propagation can be controlled by changing the isotropic medium’s refractive index. The results presented in the study by Takayama, Artigas, and Torner [17] show that these types of waves can be used as a sensing unit. It has been demonstrated in a number of publications that DSWs can exist at the interface of isotropic materials and materials with artificially designed shape anisotropy [9], [18], [19], [20], [21], [22]. Moreover, as theoretically shown in Ref. [23], [24], in the metamaterial composed of alternating layers of metals and dielectric, exotic types of surface waves such as Dyakonov plasmons and hybrid plasmons can appear. In such structures, the angular range of existence of DSWs can be extended up to

Recently, in 2019, a new type of surface waves, referred to as Dyakonov-Voigt surface waves, have been theoretically demonstrated at the interface of isotropic and uniaxial materials [25]. Unlike conventional DSWs, Dyakonov-Voigt surface waves decay is the product of a linear and an exponential function of the distance from the interface in anisotropic medium [26], [27], [28]. In contrast to DSWs, Dyakonov-Voigt surface waves propagate only in one direction in each quadrant of the interface plane.

Like in the cases of other surface waves, the feasibility of the practical use of DSWs ultimately depends on whether they can exist in resonator structures of finite size and whether they can propagate without radiative losses. In Ref. [29], it has been shown that the DSWs can be conformally transformed into the bound states of cylindrical metamaterials. Dyakonov-like surface waves have been also theoretically predicted in anisotropic cylindrical waveguides [30]. Owing to the bending of the waveguide boundary, such modes have inevitable radiative losses. In our recent work [44] we considered Dyakonov-like waveguide modes in a flat interfacial strip waveguide confined in the dimension perpendicular to the DSW propagation direction and showed that such modes can propagate without radiative losses.

This article is devoted to the theoretical study of Dyakonov-like surface states at a flat interface confined in one or two dimensions. Like in Ref. [44], we consider two anisotropic uniaxial lossless dielectrics twisted in such a way that their optical axes form an angle of

## 2 Interface of two uniaxial crystals

*x*and

*y*coordinate axes. As shown in Refs. [31], [32], [33], such a configuration supports DSWs when anisotropic media are optically positive, that is, the condition (1) is satisfied. In Refs. [34], [35], [36], [37], the problem of DSWs has been generalized to the case of two biaxial crystals. In this section, we once again describe some of the key points of DSWs in the uniaxial/uniaxial configuration which are crucial for understanding the properties of Dyakonov-like surface waves in confined media. We denote the dielectric permittivity tensor of the upper half-space as

*λ*is the wavelength and the

*z*-components of wavevectors are:

The numerical solution of the Equation (4) for the Dyakonov wave is represented in Figure 1c by the red curve. One can see that the DSW is located near the intersection of the dispersion curves of extraordinary waves with *φ* near the bisector between the crystals’ optical axes where the DSW can propagate (Figure 1b). This is in agreement with the results reported in Refs. [31], [33], [38] for the two symmetrical uniaxial anisotropic crystals.

The

To estimate the partial contributions of ordinary and extraordinary waves to the DSW, we calculate the ratio of the coefficients

*φ*and on the coordinate

*z*where the electric field is considered. For the most symmetric case (

*γ*-dependence of the DCP can be expressed analytically:

*φ*(Figure 1i) and find that the

*φ*-dependence of the DCP is weak, however for all

## 3 Reflection from boundary

Before moving on to exploring the dimensional confinement of Dyakonov-like surface states, it is important to analyze the scattering of a DSW on a single boundary perpendicular to the interface plane along which the DSW propagates. The results obtained in the previous section provide the possibility for performing this analysis.

*φ*and hitting the boundary at a varying angle of incidence

*α*(Figure 2a). In this section, we use a new coordinate system where the boundary is parallel to the

*y*-axis and the angle between the optical axes and coordinate axes equals to

*y*-component of the wavevector

*β*in the following way:

It enables us to reduce the 3D scattering problem to a 2D problem in the *XZ* plane with a fixed out-of-plane wavevector component

First, we explore the most symmetric configuration fixing the azimuthal propagation angle at *α* for the air or PEC boundaries (Figure 2c). Because both the incident and the reflected DSWs can only propagate in the limited domain near the bisector between the optical axes, the reflection of this mode without significant scattering losses can occur only at the angle *α* close to 45°. In the case of air boundary, the transmission turns to zero at

Second, we consider the case of the maximal reflection by fixing the incident angle *α* at 45° and varying the azimuthal propagation angle *φ* within the *φ*-range of DSW existence *φ* and equals to 1. In the case of the air boundary, the reflection coefficient *φ* except for the cutoff points

Figure 2e–l shows the profiles of the period average electric field intensity of the DSW being produced by the port and falling on the boundary at different incident angles *α* and at a fixed azimuthal propagation angle

We would like to emphasize once again that a nonperfect reflection at

## 4 One-dimensional confinement

Let us now study the system of upper and lower slabs, tangent to each other, confined between two parallel boundaries located at

Moreover, it also satisfies mirror boundary conditions at

*F*does not depend on

*y*-axis with the propagation constant

*n*is the mode order.

The dispersion curves of DSWMs *d*. One can see that the DSWMs appear near the intersection of the EWMs. We note that the dispersion curves of DSWMs have the cutoff points which originate from the angular cutoffs^{1}*φ*. Please note that because the orientation of optical axes of the upper and lower waveguides is fixed, the angle *φ* ultimately determines the angle of incidence *α*. Hence, each point on the DSWM dispersion curves also corresponds to a certain angle of incidence *α* close to 45°.

The waveguide width dependence of the propagation constant *d*-range where DSWM can propagate is determined by the angular existence domain of the DSW at the infinite interface. With an increase of the anisotropy factor, the existence range of DSWM broadens (Figure 3d) and, at large anisotropy, the existence ranges of DSWMs with different mode numbers *n* overlap. It is worthy to note that we compared the analytical results obtained from Eq. (15) for PEC with full-wave simulations made in COMSOL Multiphysics and observed excellent agreement (not shown in Figure 3).

Our field simulations reveal that for the first-order DSWM, there is the symmetry mismatch and the corresponding overlap integrals vanish, which indicates that the coupling of the first-order DSWM with the slabs’ waveguide modes is not possible. As there is also no radiative leakage to the air (see Figure 2f and its discussion), we conclude that the first-order DSWM has no radiative losses which results in the infinite FOM. Radiative losses of higher-order DSWM’s are fully attributed to the coupling with the slabs’ waveguide modes. We emphasize once again the importance of the obtained result that although DSWMs are generally coupled to propagating EWMs, a symmetry-protected lossless first-order DSWM exists.

*d*within the range of the DSWM’s existence are shown in Figure 4a. One can see that at

Electric and magnetic field intensity profiles of the first-order and the second-order DSWMs are shown in Figure 4b and c for the widths *d* such that the symmetry condition (17) is satisfied. For the air boundary, the electric field intensity profile of the *n*-th-order DSWM has *n* local maxima in the lower slab. Magnetic field intensity has *n* local maxima in both cases. For the PEC boundary, the situation is different: *n* (or *x* coordinate (see Supplemental Materials for details). The examples of field distribution for cases when *n* are presented in Supplemental materials.

At the end of this section, we conclude that the one-dimensional electromagnetic confinement makes DSWMs traveling along the direction where classical DSWs cannot propagate. Indeed, as is shown in Figure 1b, DSWs exist at a small angle around the bisector between optical axes of upper and lower anisotropic materials, whereas the DSWMs propagate along one of these optical axes. This feature distinguishes DSWMs from DSWs.

## 5 Two-dimensional confinement

*n*is the mode order,

*d*is the side of the square, and

*d*(at a given frequency and dielectric permittivity). The calculated by Eq. (18) values of the square side supporting DSCM are shown in Figure 5c as a function of the anisotropy factor for mode orders

We note that the structure with anisotropic rods shown in Figure 5a and b is *S*_{4}-symmetrical, that is, it is invariant under 90° rotation about the *z*-axis and subsequent mirror reflection relative to the *S*_{4}-symmetrical structures should have two singlet eigenmodes and one doublet eigenmode; ii) symmetries of these eigenmodes are determined by irreducible representations of the *S*_{4} point group. To explore this in application to our system consisting of two tangent rods surrounded by air or PEC, we simulate its eigenmodes in COMSOL. Our simulations reveal that, as expected, such a system supports waveguide modes propagating along the rods, as well as DSCMs localized at the interface between the rods. We notice that the waveguide modes can be singlets or doublets while the DSCMs are always singlets. This is due to more strict selection rules for DSCM in comparison with waveguide modes. The calculated electric and magnetic field intensity profiles of the first-order DSCM in vertical and horizontal cross-sections are shown in Figure 5f–n. One can see that in the upper (or lower) rod, the electric field is mainly localized near the boundaries *z*-projection of electric and magnetic vectors in Figure 6b. By inspecting Figure 6b, one can see that the displayed DSCMs refer to the irreducible representations either A or B of the *S*_{4} point group^{2}.

Owing to radiation losses caused by the scattering of DSWs at the air boundaries, DSCMs should have a finite *Q*-factor when rods are surrounded by air. Generally, the *Q*-factor depends on the dielectric permittivities of rods and environment, as well as on the mode order. The COMSOL simulation reveals that for the case of air boundaries and components of dielectric tensors of rods *Q*-factor of DSCMs with *n* = 1, 2, and 3 equal to 8.83, 52.98, and 197.06, respectively. Apparently, the *Q*-factor decreases with the mode order because of increasing diffraction losses caused by the violation of the total internal reflection in smaller squares. Owing to lack of scattering and absorption losses in the rods surrounded by PEC, the corresponding *Q*-factor is infinite.

Finally, DSCMs demonstrated in this section can be generalized to the case of cylinders of arbitrary rectangular shape. A DSCM in such a structure is a superposition of DSWs reflecting from boundaries at angles *α* not equal to 45° and propagating at azimuthal angles *φ* also not equal to 45°. Such a structure is no longer *S*_{4}-symmetrical, and the corresponding field distributions are less symmetrical in comparison with the case of square cylinders (See Supplemental Materials for details). Owing to the perfect reflection from the PEC boundary (Figure 2d), the *Q*-factors of DSCMs in rectangular cylinders with PEC boundaries remain infinite.

## 6 Conclusion

In conclusion, we have studied Dyakonov-like surface states which appear at the interface between two identical anisotropic dielectrics twisted in such a way that their optical axes form an angle of 90° to each other. First, we have studied the case of the infinite horizontal interface where DSWs exist in a small range of azimuthal angles. In the presence of vertical boundaries that constrain the system from two sides, electromagnetic confinement comes into play. We have demonstrated that such a one-dimensionally confined system supports DSWMs propagating along the direction where conventional DSWs do not exist. We have shown that the first-order DSWM can propagate without losses. This fact opens ample opportunities for using these modes in signal transmission lines and information processing. The existence of DSWMs can be explained in terms of the multireflection of DSWs at angles close to 45° to the interfacial strip waveguide boundaries. We have further improved this idea and considered the interface between two square cylinders made of anisotropic materials. Owing to the two-dimensional electromagnetic confinement, such a system supports Dyakonov surface cavity modes. We believe that our work can open new insights in the field of surface waves in anisotropic media, which can lead to the practical application of DSWs in optoelectronic devices.

## 7 Theoretical methods

To simulate the DSW reflection from a single boundary, we developed a model in COMSOL Multiphysics where the DSW is excited by a port plane. The field and the wavevector of the mode which are excited by the port are taken as a DSW solution at the infinite interface described in Section 2. Then, we find the *S*-parameters of such a system by calculating the fields at the reflection and the transmission sides. As a result, we obtain the total reflectance and transmittance of DSW at the boundary. We verified our numerical results obtained in COMSOL Multiphysics using the analytical solutions. When calculating models that do not have an analytical solution, we checked that the final results do not depend on the grid size and the position of the PML layers.

Authors acknowledge Ilia M. Fradkin for fruitful discussions.

Please note that the cutoff points for DSWM for the air boundary are determined approximately due to limitation of the computational domain size in COMSOL.

The difference between irreducible representations A and B in *S*_{4} point group is whether a field changes its sign under the symmetry operation *S*_{4}. See [43] for details.

^{}

**Author contribution:** All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission. D.C. conducted COMSOL simulations. E.A. developed a theoretical model, S.D. conceived the idea of this manuscript and organized the manuscript preparation, N.G. supervised the project and provided the valuable advice on the manuscript.

^{}

**Research funding:** This work was supported by the Russian Foundation for Basic Research (Grant No. 18-29-20032).

^{}

**Conflict of interest statement:** The authors declare no conflicts of interest regarding this article.

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## Footnotes

## Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2020-0459).

## Footnotes

^{1}

Please note that the cutoff points for DSWM for the air boundary are determined approximately due to limitation of the computational domain size in COMSOL.

^{2}

The difference between irreducible representations A and B in *S*_{4} point group is whether a field changes its sign under the symmetry operation *S*_{4}. See [43] for details.