Corresponding author: Valery R. Sobol (vrsobol@mail.ru)

Dispersion relations for a transverse electromagnetic wave in a hypothetical medium having a low crystalline symmetry and double anisotropy are formulated phenomenological based on the general concepts of Maxwell's equations. Formalism for electrical and magnetic ordering of a medium is represented by tensor coefficients of the second rank in a general coordinate system corresponding to an external problem of incidence, reflection and refraction of a transverse wave at the interface. Based on the obtained dispersion law, the Snell law of refraction for the wave vector is displayed, as well as, some special cases for a section of a wave surface and a ray surface in the plane of incidence. Following Fresnel approach as usual, two characteristic types of linear polarization of an electric induction vector of a propagating wave are considered for a wave entering a double anisotropic medium at an arbitrary angle to the interface. Based on a character of a cross-section of a wave surface and of a ray surface for the plane of incidence, the degree of influence of anisotropy on a propagating field is discussed in terms of external/internal conic refraction of ordinary/extraordinary waves in a transparent medium. Following to initially general form of the permittivity

At present the validity of standard retrieval methods that assign bulk electromagnetic properties, such as a electric permittivity ε and a magnetic permeability μ, from calculations of the scattering (

The electromagnetic properties of an inhomogeneous composite can be determined exactly by solving Maxwell’s equations, which relate a local electric and magnetic fields to the local charge and current densities. When the particular details of the inhomogeneous structure are unimportant to the behavior of relevant fields of interest, the local field, charge, and current distributions are averaged, yielding a macroscopic form of Maxwell’s equations [

Here we discuss a case of homogeneous material on a base of some hypothetical anisotropic medium of crystalline type which has macroscopically ordered properties both of electric and magnetic nature. Electrical ordering in optics in some crystal media is widely known to lead to birefringence (for example crystalline quartz, calcite, aragonite, etc.). It is reasonably to propose that a magnetic ordering also can stimulate some similar behavior of a linearly polarized electromagnetic wave via refraction index that is via a magnetic permeability.

In media having a double type of anisotropy, for example in so called transparent ferroelectrics, it is rather difficult to pose a question of an experimental solution of inverse Fresnel problem that is to restore the properties of a medium based on a reflection experiment. Such medium may have two systems of optical axes of electric and magnetic nature, which as usually said in general do not necessarily coincide in space. Accordingly, it is rather difficult to describe the features of a reflection of polarized light in the form of Fresnel relations. Moreover, it should be noted that even an attempt to describe a reflection of polarized light from an interface with medium having usual type of anisotropy (for example mentioned calcite, aragonite or crystalline quartz etc.) is in a general case far from a trivial task namely for a wave being polarized corresponding to extraordinary type (when a refractive index is a function of propagate angle). In this Fresnel problem, the optical axes are needed to be correlated with an external coordinate system that determines the plane of incidence, angles of incidence, and refraction. But, in fact, for an external task, which is the search for Fresnel coefficients, a main aim is dispersion law, the knowledge of the orientation of a main coordinate axes is not so relevant against a background of correct matching of the field components for incident, reflected and refracted waves. That is, on the contrary, about a medium itself, its type and a nature of its axes, it is quite possible to draw some conclusions based on a recording and analysis of a dispersion law and corresponding Fresnel relations, with additional use of results from measuring of reflection coefficients at accepted refractive conditions on interface.

Within of the mentioned above, a purpose of this work is to express a dispersion law for transverse electro-magnetic wave at two typical linear polarization for electric induction vector with respect to a selected plane, being a plane of incidence. Here we are talking about two Fresnel waves propagation in homogeneous crystalline medium being ordered as electrically as magnetically. One of mentioned wave is linearly polarized in a plane of incidence, and another wave is linearly polarized normally to the plane of incidence.

In this representation, the dispersion law, after writing it down, allows us to formulate the law of refraction for wave vector. Respectively there appears a possibility to obtain an expressions required subsequently to formulate Fresnel incidence/reflection relations at an interface. So a nature of linearly polarized wave incidence/reflection/refraction/propagation it makes sense to formulate and describe via initial dispersion relations in a general coordinate system of dielectric and magnetic tensors. At first we do not use a coordinate system of principal axes but use a general coordinate system of tensor coefficients, when one of a tensor coordinate planes coincides with a plane of incidence of a wave and another plane is that of an interface. As a result, based on a written dispersion law, it will be possible to identify the most rational approaches to experimentally solving an inverse Fresnel problem. That is within the framework of achieved relations and based on the results of measuring a reflection coefficient to restore a type and parameters of tensor material constants.

The study of Fresnel reflection, as mentioned, is promising for a double anisotropy research in view of a possibility of identifying magnetic and optical properties based on the results of inverse consideration of a dependence of the measured reflection coefficient and a state of polarization as a function of an angle of incidence. As previously stated, the results of a development of synthesis technologies, materials scientists are successfully mastering a class of new substances being complex in composition and crystal structure. These materials are usually ordered by both electrical and magnetic degrees of freedom and include transparent thin layers of ferromagnets, magnetic semiconductors-ferrites, materials of a so-called ferroelectric segment [

In given case, an optical medium having a double type of anisotropy is considered using an example of a model ferroelectric structure having a low level of crystal symmetry, for example, of a rhombohedral type and lower. Here we consider a phenomenological aspect and leave the question of microscopic mechanisms of a formation of double ordering. Does ordering appear due to direct positive/negative exchange, or due to indirect exchange/over exchange of magnetic ions in the crystal structure, or due to other mechanisms. We simply postulate that an optically transparent medium can be characterized by orthogonal polar Hermitian tensors of electric permittivity

As mentioned above, the crystalline medium in a general coordinate system of accepted dielectric and magnetic tensors

The known relationships between the components of the vectors of electrical and magnetic induction (

_{0}^{–1}^{–1}

_{0}^{–1}^{–1}

here ε_{0} and μ_{0} are an electric and magnetic constants, symbols like ^{–1} etc. – denote inverse tensors/their components.

The geometry of a problem, presented in a tensor coordinate system, is represented for a linearly polarized wave as it enters a medium according to an incidence-reflection-refraction scenario (Fig.

The geometry of an entry of a linearly polarized wave (angle of incidence – φ, angle of refraction – ψ) into the surface (

In Fig.

_{x}_{y}_{z}

The condition of continuity of the normal components of an electrical induction vector at an interface leads to fact that when a transverse electromagnetic wave falls from an isotropic medium onto the interface with an anisotropic medium, the tangential component of a wave vector remains continuous and a wave vector in a _{x}_{y}_{z}

To formulate the relation for a dispersion expression of a refracted wave, we use Maxwell’s equations and respective tensor equation (5)

In a basis of unit vectors

_{x}_{y}_{z}^{і}^{(ω}^{t}^{kr}^{)}, (6)

_{x}x_{y}y_{z}z

Here we emphasize for clarity that _{x}_{y}_{z}

After some detalization (10) is transformed to (11)

here, by coinciding indices, is accepted, as usual, summation.

Next steps leads to (12)

Such for a component of _{x}

In (13) there presented a most general case of transverse wave propagation through crystalline medium of a type (1). Here a wave vector _{x}_{y}_{z}_{x}_{y}_{z}

First line (14) is

Second line (15) is

Third line (16) is

It is obvious that in general, this system of equations (14)–(16) is somewhat redundant in terms of a number of elements. Nevertheless, this system (14)–(16) is extremely attractive in the sense that it allows one to obtain dispersion relations for any directions of a wave vector at any linear polarization. This means that using this system in a future it will be possible to construct a topology of a wave vectors surface in explicit form. In addition, it is possible, as will be shown below, to obtain the Snell relations, which relate the angles of incidence and refraction for a wave vector for mentioned type linear polarization.

From the point of view of the generality of the representation, it was important namely to identify the initial principles of the genesis of properties in the formalism of optical axes both electric and magnetic nature at arbitrary orientation of the intrinsic tensors ε and μ both before and after their diagonalization in relation to the external coordinate system in the reflection/refraction problem. The analysis provided an understanding of the situation and the requirement for the crystal that at least the main cross section of one of the tensors be in the plane of incidence. This can be used in reflectance spectrophotometry when solving the inverse problem of identifying the anisotropy parameters of complex media of the indicated type.

When specifying a wave vector in one of the coordinate planes, in accepted case it is a

The mentioned dispersion law is understood here as a form of a wave vector as a function of frequency, parameters of tensor coefficients and an angle of propagation, or more precisely, refraction. In fact, when specifying the components of a induction vector

For example, for a simplest case of a linearly polarized wave in a medium being isotropic in optical properties, system (14)–(16) degenerates to determinant (17), which leads to well-known dispersion equation (18) for two types of polarization both for a plane of incidence and normal to it

Indeed, for an isotropic medium, the tensors degenerate to a spherical form, turn into scalars, and the dispersion law corresponds to a well-known expression where all diagonal components of tensors

Essentially, a carried out limiting verification of system (14)–(16) indicates the correctness of its representation. This gives grounds to use these equations to represent a dispersion law also for anisotropy of the properties of the medium. Accordingly, for a chosen anisotropy of a most general type for a polarization of a vector

For a wave polarized in a plane of incidence (

In both cases considered (19), (20), a refractive index is a function of a propagation angle ψ. In terms of generally accepted expressions, this corresponds to condition that both waves are unusual when they propagate in a

It should be noted that the dispersion law and a condition of continuity of the tangential components of a wave vectors at an interface make it possible to write relations for Snell’s law and express an angle of incidence as a function of the angle of refraction. For example, for polarization of a wave in a

Here ε_{1} and μ_{1} are electric permittivity and magnetic permeability of upper isotropic medium (for example, of vacuum).

Figure

As seen (see Fig.

An example of Snell's refraction law for a usual wave and an extraordinary wave being polarized by vector _{xx}_{yy}_{zz}_{xx}_{yy}_{zz}_{ik}_{ik}_{xx}_{yy}_{zz}_{xx}_{yy}_{zz}_{ik}_{ik}_{xx}_{yy}_{zz}_{xx}_{yy}_{zz}_{ik}_{ik}_{xx}_{yy}_{zz}_{xx}_{yy}_{zz}_{ik}_{ik}

Unfortunately, the relationships between the angles of incidence and refraction do not allow us to specify the details of a spatial dependence of a wave vector. But expressions (19) and (20) themselves make it possible to graphically display a topology of a wave vector in a plane of incidence. In other words, (19) and (20) allow to construct a section of a surface of wave vectors in one of a coordinate planes (of incidence-reflection-refraction) and not in a system of the main axes of tensors as it was discussed in the works of Fresnel. Here we are talking about a coordinate system corresponding to an external problem of refraction/reflection at an interface. In essence, here a wave vector, expressed only through the components of these tensors determines a topology of a spatial distribution of an effective refractive index in a plane of incidence for a wave of corresponding type of polarization. Respectively a section of a ray surface can be realized according to a well-known procedure with a section of the wave surface by replacing in expression for _{ik}^{-1} and μ_{ik}^{-1} onto their inverses. That is, by representing of _{ik}_{ik}_{ik}^{-1}, μ_{ik}^{-1} there have been got a section of a ray surface

Actually, according to the general concepts, the wave vector in the analysis is displayed as a function of the inverse components of the tensors ε_{ik}^{-1}, μ_{ik}^{-1} and not as ε_{ik}_{ik}

Both types of sections, that is, _{xy}_{z}_{xy}_{z}

A type of effective cross section in terms of an effective refractive index for a wave surface _{xy}_{xy}_{z}_{z}_{xx}_{yy}_{zz}_{xy}_{yx}_{xz}_{zy}_{zx}_{yz}_{xx}_{yy}_{zz}_{xy}_{yx}_{xz}_{zy}_{zx}_{yz}

As can be seen (see Fig.

This question can be answered after bringing the tensors from Fig.

In this case, the orientation of unit vectors of the new main axes

For the reduced tensor _{m}_{m}_{m}

Accordingly, without any geometric constructions, it is clear that a main sections in a system of reduced axes _{m}_{m}_{m}

Before nowadays, in classical works only an electrically anisotropic optical medium was discussed. After reducing its tensor

Based on this analysis it is clear that a coincidence of phase velocities along a direction of a bi-normals type is realized for polarization not only for main planes, but also outside them. This exactly correlates with a concept of conical refraction, which will be discussed below.

Once more an example of double anisotropy represents a phenomenon of conical refraction more clearly in a following Fig.

ε_{xx}_{yy}_{zz}_{xy}_{yx}

ε_{xz}_{zy}_{zx}_{yz}_{xx}_{yy}_{zz}

μ_{xy}_{yx}_{xz}_{zy}_{zx}_{yz}

Here as mentioned an electric tensor is the same as before (see Fig.

Such in Fig.

The type of effective cross section in terms of an effective refractive index for a wave surface _{xy}_{xy}_{z}_{z}

On the other hand, a beam propagating in isotropic medium at a small angle to an

Further it is a reason to show a view of a section for wave vector to the power 2 in terms effective refractive index. So Fig.

The topology of a cross section for a wave vector to the power 2. A square of effective refractive index for waves entering a double anisotropic optical medium when these are polarized in a plane of incidence (_{xx}_{yy}_{zz}_{xx}_{yy}_{zz}_{ik}_{ik}

A stronger elongation of a wave section for

It is important that if a coordinate axes are chosen so that components of type ε_{iz}_{zi}_{xy}_{yx}

In a more particular case, for example, if all diagonal components in (26) are equal and if this tensor is symmetric, which initially should be assumed if there is no external influence, it is obvious that an expression in question (26) can be reduced to the form (27). This means that as before in a plane of incidence (

Naturally, if the tensor _{xy}_{yx}

For a wave being polarized in an incidence plane

Next, it makes sense to consider once more particular case of anisotropy within an accepted coordinate system (

a wave being

The form of dispersion law (30) corresponds to a more rational orientation of the crystalline axes and reflective surface of a medium in relation to the coordinate system of tensor coefficients that is to more benefit sample surface orientation in an external problem of an incidence-reflection. The rationality of this representation means that with a dielectric tensor of a form (29), a wave vector of the incident and refracted waves are placed in a main plane of tensor (

For example for an anisotropy of a type (31) a dielectric constant tensor after a reducing it to the main axes corresponds to the expression (32).

ε_{xx}_{yy}_{zz}_{xy}_{yx}

ε_{xz}_{zy}_{zx}_{yz}_{xx}_{yy}_{zz}

μ_{xy}_{yx}_{ik}

Respectively, the orientation of the unit vectors of new main axes

The relation (33) indicates that choosing a rational tensor coordinate system that coincides with a coordinate system of an external reflection problem one can represent a section of a wave surface in an incidence plane (

The section of a wave surface and of a ray surface in plane of incidence/reflection (_{xy}_{z}_{xy}_{z}_{xy}_{z}

Following Fig. _{zz}_{xy}

View of one part of a wave surface for a wave being linear polarized in a plane of incidence. An electrically anisotropic medium has a parameters of a type (31), a wave corresponds to an extraordinary that. Range variation of angles is: a refraction (latitude) angle, ψ is: 0 – π (placed in a plane of Fig.

Note that as mentioned above in a general case a direction of an energy flux density vector

Thus, for the case of a particular type of dispersion law of type (29), a bi-anisotropic medium can be represented in terms of familiar concepts. In the accepted approximation, a plane of a main section coincides with the

The formulated dispersion relations make it possible to write Snell's law for the wave vector and, in a future, to obtain a Fresnel ratio for amplitudes for electric intensity vectors, namely a ratio of a reflected wave amplitude to an incident that. Relationships for amplitudes are available for implementation based on standard continuity conditions.

Returning to the most general form of double-type anisotropy, we can conclude that an arbitrary choice of a common coordinate system for tensors means a more complicated presentation of two extraordinary waves with polarization in a plane of incidence and perpendicular to it. The graphical construction of a topology of a wave surface section in a

The Maxwell's equations in presentation of a coordinate system of an external problem of a reflection/refraction for a transverse electric-magnetic wave allows anybody to write down a wave dispersion law after refraction of a polarized wave into an anisotropic medium being ordered by both electrical and magnetic properties. A double type of ordering it makes sense to represent in a components of a polar tensors for electric and magnetic permeability taking into account a fact that these tensors must be symmetrical at the absence of medium own optical activity or of external influence.

Tensor equations for a wave vector being presented in coordinate form allow to obtain a dispersion equations for a wave vector and a ray vector at an interface in a plane of incidence being that of accepted tensor coordinate system as well as for other intermediate planes of incidence and to construct wave vector surface. The coupling relations for wave vectors at an interface allow by using a Snell refraction law to generalize Fresnel relations in future for reflection coefficients.

The dispersion law in a bi-anisotropic medium for a transverse wave for two types of polarization of an electric induction vector is suitable for constructing of a wave surface of a refracted wave with subsequent identification of the features of a medium by a number of optical axes, by their nature, by a type of wave refraction and by a directions of phase and group velocities.

Coordinate representation of dispersion law confirms an expediency of organizing of reflection experiment while ensuring a rational geometry of the problem. This implies an orientation of a crystalline symmetry axis of higher order into a plane of incidence. As result it make it possible to more rationally use empirical data on reflection when reconstructing the features of a crystalline medium due to the reduction in a number of tensors components during the solution of an inverse refraction/reflection problem.

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