Birefringence

## Abstract

The title “Dyakonov surface waves: anisotropy enabling confinement on the edge” plainly sets the scope for this chapter. The focus here is on the formation of bounded waves at the interface of two distinct media, at least one of them exhibiting optical anisotropy, which are coined as Dyakonov surface waves (DSWs) in recognition to the physicist who reported their existence for the first time. First, the general aspects of the topic are discussed. It also treats the characterization of bounded waves in isotropic-uniaxial multilayered structures, allowing not only the derivation of the dispersion equation of DSWs but also that of surface plasmons polaritons (SPPs), for instance. Furthermore, the interaction of such surfaces waves, with the possibility of including guided waves in a given planar layer and external sources mimicking experimental setups, can be accounted for by using the transfer matrix formalism introduced here. Finally, special attention is devoted to hyperbolic media with indefinite anisotropy-enabling hybridized scenarios integrating the prototypical DSWs and SPPs.

### Keywords

- surface waves
- anisotropy
- transfer matrix formulation

## 1. Introduction

The planar interface of two dissimilar materials plays a relevant role in many optical phenomena. In recent years, particularly, evanescent waves have been used in newly developing technologies such as near-field spectroscopy. The electromagnetic surface wave, which is intimately tied to the interface, travels in a direction parallel to the interface but, on either side of the interface, its amplitude is imperceptible after a certain distance from the interface. The notion of an electromagnetic surface wave made a significant appearance in 1907 when Zen-neck [1] authored a theoretical paper exploring the possibility of a wave guided by the interface of the atmosphere and either Earth or a large body of water. His focus was on radio waves, a region of the electromagnetic spectrum far from the optical regime in which we are particularly interested, but the principles involved are the same, owing to the scale invariance of the Maxwell postulates.

Yet, nearly a century later, a unique type of wave, the surface plasmon polariton (SPP) wave, that dominates the nanotechnology scene, at least at optical frequencies, resulted in wonderful developments with the creation of extremely sensitive bio/chemical sensors, and improvements in this mature technology continue to this day [2]. Even in this highly developed application, the two partnering materials which meet at the interface may be simple: one is a typical metal, a plasmonic material at optical frequencies, and the other is a homogeneous, isotropic, dielectric material. While the interface of a plasmonic material and a polarizable material supports SPPs, a variety of other types of surface waves can be supported by the interface of two polarizable materials. Since polarizable materials such as dielectric materials are less dissipative, in general, than plasmonic materials such as metals, the advantage of these materials for long-range propagation of surface waves is apparent.

The interface of two homogeneous dielectric materials of which at least one is anisotropic may support surface-wave propagation of another type, even though the real parts of all components of the permittivity dyadics of both materials are positive. Interest in surface waves guided by the interface of two dielectric materials began to take after Dyakonov in 1988 [3] explored the propagation of a surface wave guided by the interface of a uniaxial dielectric material and an isotropic dielectric material. The Dyakonov surface wave (DSWs) is the focus of this chapter.

In this chapter, we perform a thorough analysis of DSWs taking place in semi-infinite anisotropic media. Basic concepts related to the propagation of electromagnetic waves in homogeneous media will be introduced, including isotropic and anisotropic materials. Birefringent metal-dielectric (MD) lattices will be also considered as a contribution of meta-materials in the development of DSWs [4]. Special emphasis will be put when the effective-medium approaches induce unsatisfactory results, which occur in most experimental configurations. Practical cases will be analyzed including dissipative effects due to Ohmic losses of the metal.

## 2. Wave propagation in bulk media and interfaces

In this section we introduce the basic concepts related to the propagation of electromagnetic waves in homogeneous media, including isotropic and anisotropic materials. We describe in detail complex multilayered structures. For that purpose, we introduce a transfer matrix formulation that applies to isotropic and uniaxial media simultaneously. Finally, we discuss the conditions that give rise to surface waves at the interface of two isotropic media; the case of dealing with anisotropic media is considered in Section 5. Moreover, we obtain the dispersion equation for SPPs, which appears at the interface between a dielectric and a metal.

### 2.1. Wave propagation in isotropic media

In this section, we consider the propagation of electromagnetic waves in linear, homogeneous, and isotropic dielectrics. Under these conditions, the relative permittivity ϵ relating **E** and **D** is a scalar constant. Considering that the medium is free of electric charges and currents, and taking into account the medium equation

Now, each of the scalar components of

When the electromagnetic wave is plane and monochromatic, all components of the electric and magnetic fields are harmonic functions in time and space at the time frequency

where

We point out that

### 2.2. Wave propagation in uniaxial media

Uniaxial crystals are media with certain symmetries that make them have two equal principal refractive indices:

Birefringent material | |||
---|---|---|---|

Crystal quartz | 1.547 | 1.556 | 0.009 |

MgF | 1.3786 | 1.3904 | 0.0118 |

YVO_{4} | 1.9929 | 2.2154 | 0.2225 |

Rutile (TiO_{2}) | 2.65 | 2.95 | 0.3 |

E7 liquid crystal | 1.520 | 1.725 | 0.205 |

Calomel (Hg_{2}Cl_{2}) | 1.96 | 2.62 | 0.68 |

Let us now consider the propagation of wave planes in uniaxial media. To obtain the eigenvalues associated with plane-wave propagation, we proceed in a similar way as in Section 2.1 for isotropic media, but taking into account that now the relative permittivity

In this case, the electric field is proportional to the vector

### 2.3. Matrix formulation for multilayered media

In this section we look at the case of multilayered media composed of different nonmagnetic materials which are separated by planar parallel interfaces, displaced on *i*”, which lies in

#### 2.3.1. Electromagnetic fields in a uniaxial elementary layer

Let us first consider a multilayered media of uniaxial materials. For simplicity, let us take into account only relative permittivities of the form

and for extraordinary waves (here

taken from Eq. (4).

The total electric field of the elementary layer “

The part of the electric field that varies along with the spatial coordinate

where the amplitude

In the case of counter-propagating waves, we take into consideration that

The vector fields

For convenience, the field function

The complete set of amplitudes is

For completeness we calculate the magnetic field in a given elementary layer “

In the previous equations, the new vector fields are set as

We point out that the vector fields

#### 2.3.2. Boundary conditions for anisotropic layered media

Once we have a complete description of the wave fields in every elementary layer of our metamaterial, we have to impose some boundary conditions at the interfaces

We defined the following matrix, explicitly given as

On the other hand, we introduced the amplitude column vectors

The matrix formulation can also be used to relate the amplitude vector

being

where

#### 2.3.3. Electromagnetic fields in layered isotropic media

At this point, once we have described the electromagnetic fields in uniaxial media, let us study a multilayered media composed of isotropic materials. Considering an isotropic medium of relative permittivity

Formally,

Note that all these amplitudes have zero dephase at

In Eq. (22a) and (22c) we have included the field vectors

In the previous equation we introduced the vector fields

Note that

#### 2.3.4. Application of the boundary conditions

At a given interface

In the previous matrix equation, we introduced the element

Finally, the amplitude vectors now are represented as

The matrix formulation can also be used to relate the amplitude vector

which takes into account the amplitude phase shift due the finite width

## 3. Surface modes in isotropic media

The main purpose of this chapter is the analysis of Dyakonov surface waves, which originally was formulated for an isotropic medium and a uniaxial crystal. However, this analysis is developed in Section 5. Here we introduce the most well-known surface waves arisen at the interface between isotropic media of different dielectric constants. In addition, these surface waves will play a relevant role when dealing with metal-dielectric multilayered structures.

The so-called surface plasmon polaritons are waves that propagate along the surface of a conductor, usually a metal [2, 8, 9]. These are essentially light waves that are trapped on the surface, evanescently confined in the perpendicular direction and caused by their interaction with the free electrons of the conductor, the latter oscillating in resonance with the electromagnetic field. To describe these wave fields, we use the matrix formalism applied in the vicinity of a single interface between two isotropic media with different dielectric permittivities.

Let us consider the propagation of bound waves on the interface between two semi-infinite media, which are denoted as medium 1 and medium 2 with dielectric permittivities

As we are only interested in bound states; the elements of the remaining field vectors read as

The values of the amplitudes

To accomplish Eq. (30a), the following equation must be satisfied,

where

At

## 4. Multilayered plasmonic lattices

Wave propagation in periodic media can be treated as the motion of electrons in crystalline solids. In fact, formulation of the Kronig-Penney model used in the elementary band theory of solids is mathematically identical to that of the electromagnetic radiation in periodic layered media. Thus, some of the physical concepts used in material physics such as Bloch waves, Brillouin zones, and forbidden bands can also be used here. A periodic layered medium is equivalent to a one-dimensional lattice that is invariant under lattice translation.

Here we will treat the propagation of electromagnetic radiation in a simple periodic layered medium that consists of alternating layers of transparent nonmagnetic materials with different electric permittivities. The layers are set in a way that the

According to the Floquet theorem, solutions of the wave equation for a periodic medium may be set in the form

represents the dispersion relation for TE and TM modes, written in a compact way. Representing

we find the dispersion equation of a binary periodic medium for each polarization.

Neglecting losses in the materials, regimes where

In Figure 3, the transverse wavenumber reads as

For ultra-thin metallic layers, for instance,

### 4.1. Effective medium approach

For near-infrared and visible wavelengths, nanolayered metal-dielectric compounds enable a simplified description of the medium by using the long-wavelength approximation, which involves a homogenization of the structured metamaterial [10, 11]. The effective medium approach (EMA), as Rytov exposed in his seminal paper [12], involves representing MD multilayered metamaterial as an uniaxial plasmonic crystal, whose optical axis is normal to the layers (in our case, the x-axis is the optical axis), a procedure that requires the metallic elements to have a size of a few nanometers. This is caused by the fact that transparency of noble metals is restrained to a propagation distance not surpassing the metal skin depth. In Ref. to this point, recent development of nanofabrication technology makes it possible to create such subwavelength structures. Under this condition, the plasmonic lattice behaves as a uniaxial crystal characterized by a relative permittivity tensor

gives the permittivity along the optical axis, and

being now

The validity of the EMA is related on the assumption that the period

### 4.2. Hyperbolic media

As we have seen in Section 3.3, nanolayered metal-dielectric compounds behave like plasmonic crystals enabling a simplified description of the medium by using the long-wavelength approximation [10, 11, 12]. Under certain conditions, the permittivity of the medium set in the form of a second-rank tensor includes elements of opposite signs, leading to a metamaterial of extreme anisotropy [13, 14]. This class of nanostructured media with hyperbolic dispersion is promising metamaterials with a plethora of practical applications from biosensing to fluorescence engineering [15].

Type I hyperbolic media refers to a special kind of uniaxially anisotropic media, that can be described by a permittivity tensor where element

The system under analysis is a periodic binary medium, where we take as medium 1 a transparent dielectric medium that is ideally nondispersive. In our three numerical simulations we take a lossless Drude metal where its permittivity is

In Figure 4(a), we represent the permittivities

Furthermore, the hyperbolic dispersion exists up to a frequency

for which

leading to the so-called canalization regime. In general,

## 5. Dyakonov surface waves

Dyakonov surface waves (DSWs) are another kind of surface waves, supported at the interface between an optically isotropic medium and a uniaxial-birefringent material. In the original work by Dyakonov (English version was reported in 1988 [3]), the optical axis of the uniaxial medium was assumed in-plane with respect to the interface. This is the case we deal with here.

The importance of DSWs for integrated optical applications, such as sensing and nano-waveguiding, was appreciated in a series of papers [17, 18]. Indeed Dyakonov-like surface waves also emerge in the case that a biaxial crystal [19] or a structurally chiral material [20] takes the place of the uniaxial medium. The case of metal-dielectric (MD) multilayers as structurally anisotropic media is especially convenient since small-filling fractions of the metallic inclusions enable metamaterials with an enormous birefringence, thus enhancing density of DSWs and relaxing their prominent directivity [21, 22, 23].

### 5.1. Dispersion equation of DSWs

The system under study is the plotted in Figure 5, where we have two semi-infinite media, one of them is isotropic and the second one is an MD lattice. In our case, the indices “1” and “2” make reference to the plasmonic lattice and the isotropic medium, respectively. We have previously reported a comprehensive analysis of this case in [4]. As we have seen earlier, the plasmonic lattice can be taken as an effective uniaxial crystal. In this case, the permittivity along its optical axis,

Since we treat the plasmonic lattice as a uniaxial crystal, we may establish analytically the diffraction equation that gives the 2D wave vector

Moreover, these fields are evanescent in the isotropic medium and in the superlattice. In the anisotropic medium (

where the ordinary and extraordinary waves in the effective uniaxial medium decay exponentially with rates given by

where the evanescent decay for TE and TM modes is

Once we have the amplitudes in both sides of the interface, we apply the boundary conditions at

where the transmission matrix *hybrid* polarization modes. Using the elements

Eq. (43) can be rewritten as a set of two independent matrix equations, namely

Note that *anisotropic* medium.

Dyakonov equation is obtained by means of letting the determinant of

which provides a spectral map of allowed values

Assuming that

### 5.2. DSWs in nano-engineered materials

To illustrate the difference between using conventional birefringent materials and plasmonic crystals, we solve Eq. (47) for liquid crystal E7 with

Consequently, the solution for Eq. (47) can be traced near the curves

#### 5.2.1. Nonlocal effects

As we discussed in Section 4.1, the EMA is limited to metallic slabs’ width

We emphasize that moderate changes in the birefringence of the plasmonic crystal will substantially affect the existence of DSWs. More specifically, an enlargement of

We conclude that, in order to excite DSWs, one may counterbalance the decrease of birefringence in the plasmonic lattice by means of a dielectric substrate of higher index of refraction. To illustrate this matter, the dispersion Eq. (47) for DSWs is represented in Figure 7, in addition to using the values of

For an N-BAK1 substrate, as shown in Figure 7(a),

#### 5.2.2. Dissipative effects

Up to now, we have avoided another important aspect of plasmonic devices namely dissipation in metallic elements. In this regard, effective permittivities are fundamentally complex, and consequently the Dyakonov Eq. (47) is expected to give complex values of

Figure 8(a) depicts the dispersion curve corresponding to dissipative DSWs, for the case of a plasmonic MD lattice with

We observe that the dispersion curve for dissipative DSWs is flatter and larger than the curve obtained by neglecting losses. Specifically

### 5.3. New families of DSWs in lossy media

In this section we carry out a thorough analysis of DSWs that takes place in lossy uniaxial metamaterials. Special emphasis is put when the effective-medium approach induces satisfactory results. The introduction of losses leads to a transformation of the isofrequency curves, which deviates from spheres and ellipsoids, as commonly considered by ordinary and extraordinary waves, respectively. As a consequence, one can find two different families of surface waves as reported by Sorni et al. [25]. One family of surface waves is directly related with the well-known solutions derived by Dyakonov [3]. Importantly, the existence of a new family of surface waves is revealed, closely connected to the presence of losses in the uniaxial effective crystal. We point out that the solutions to Dyakonov equation presented earlier are partial ones insofar as the

## 6. Dyakonov surface waves in hyperbolic media

In this section we perform a thorough analysis of DSWs taking place in semi-infinite MD lattices exhibiting hyperbolic dispersion. Part of this section was previously reported by Zapata-Rodríguez et al. [26]; we point out that recently further studies on DSW in hyperbolic metamaterials have been reported by other authors [27]. Our approach puts emphasis on the EMA. Under these conditions, different regimes can be found including DSWs with nonhyperbolic dispersion. The system under analysis is again as depicted in Figure 5. For simplicity, we assume that dielectric materials are nondispersive; indeed, we set

In the special case of the surface wave propagation perpendicular to the optical axis (

which resembles the dispersion equation of conventional SPPs [see Eq. (31)]. Here we have purely

Next we describe a specific configuration governing DSWs, subject to a low value of the refractive index

Note that

In Figure 9(a) and (b), we illustrate the dispersion equation of DSWs for two different frequencies within the spectral range

being

In the high-frequency band

### 6.1. DSWs in band-gap hyperbolic media

In previous sections we demonstrated that the presence of metallic nano-elements leads to nonlocal effects and dissipation effects which reshape the propagation dynamics of the surface signal. Here, we briefly discuss the extraordinary favorable conditions which may appear in band-gap metal–insulator-layered media for the existence of DSWs. As reported thoroughly by Miret et al. [29], engineering secondary bands by tuning the plasmonic-crystal geometry may lead to a controlled optical anisotropy, which is markedly dissimilar to the prescribed hyperbolic regime that is derived by the EMA, however, assisting the presence of DSWs on the interface between such hyperbolic metamaterial and an insulator.

In particular, a surface wave propagating on an Ag-Ge grating was considered, where the environment medium that is set above the metallic grating is formed by

Considering now a realistic nanostructure consisting of Ag layers of

## 7. Summary

In this chapter we provide several methods to analytically calculate and numerically simulate modal propagation of DSWs governed by material anisotropy. We focused on the spatial properties of DSWs at optical and telecom wavelengths, particularly using uniaxial metamaterials formed of dielectric and metallic nanolayers. We developed an electromagnetic matrix procedure enabling different aspects reviewed in this chapter, specially adapted to complex multilayered configurations. The EMA results are particularly appropriate for the characterization of the form birefringence of a multilayered nanostructure, though limitations driven by the layers width have been discussed. Through a rigorous full-wave analysis, we showed that hybrid-polarized surface waves may propagate obliquely at the boundary between a plasmonic bilayer superlattice and an isotropic loss-free material. We revealed that realistic widths of the slabs might lead to solutions which deviate significantly from the results derived directly from the EMA and Dyakonov analysis. Finally, we showed that excitation of DSWs at the boundary of an isotropic dielectric and a hyperbolic metamaterial enables a distinct regime of propagation. It is important to note that the properties of the resulting bound states change drastically with the index of refraction of the surrounding medium, suggesting potential applications in chemical and biological sensing and nanoimaging.

This chapter was supported by the Qatar National Research Fund (Grant No. NPRP 8-028-1-001) and the Spanish Ministry of Economy and Competitiveness (Grants No. TEC2014-53727-C2-1-R and TEC2017-86102-C2-1R).