A Operator of evolution

Remarkable optics of short-pitch deformed helix ferroelectric liquid crystals: symmetries, exceptional points and polarization-resolved angular patterns

Abstract

In order to explore electric-field-induced transformations of polarization singularities in the polarization-resolved angular (conoscopic) patterns emerging after deformed helix ferroelectric liquid crystal (DHFLC) cells with subwavelength helix pitch, we combine the transfer matrix formalism with the results for the effective dielectric tensor of biaxial FLCs evaluated using an improved technique of averaging over distorted helical structures. Within the framework of the transfer matrix method, we deduce a number of symmetry relations and show that the symmetry axis of lines (curves of linear polarization) is directed along the major in-plane optical axis which rotates under the action of the electric field. When the angle between this axis and the polarization plane of incident linearly polarized light is above its critical value, the points (points of circular polarization) appear in the form of symmetrically arranged chains of densely packed star-monstar pairs. We also emphasize the role of phase singularities of a different kind and discuss the enhanced electro-optic response of DHFLCs near the exceptional point where the condition of zero-field isotropy is fulfilled.

ferroelectric liquid crystal; transfer matrix method; polarization of light; polarization singularities; exceptional points; polarization-resolved conoscopic pattern
pacs:
61.30.Gd, 78.20.Jq,77.84.Nh,42.70.Df, 42.25.Ja

I Introduction

Over the last more than three decades ferroelectric liquid crystals (FLCs) have attracted considerable attention as promising chiral liquid crystal materials for applications in fast switching display devices (a detailed description of FLCs can be found, e.g., in monographs Lagerwall (1999); Oswald and Pieranski (2006)). Equilibrium orientational structures in FLCs are represented by helical twisting patterns where FLC molecules align on average along a local unit director

(1)

where is the smectic tilt angle; is the twisting axis normal to the smectic layers and is the -director. The FLC director (1) lies on the smectic cone depicted in Fig. a with the smectic tilt angle and rotates in a helical fashion about a uniform twisting axis forming the FLC helix with the helix pitch, . This rotation is described by the azimuthal angle around the cone that specifies orientation of the -director in the plane perpendicular to and depends on the dimensionless coordinate along the twisting axis

(2)

where is the helix twist wave number.

(a) Smectic cone
(b) Planar aligned FLC cell
Figure 1: (Color online) Geometry of (a) smectic cone and (b) planar aligned FLC cell with uniform lying helix.

The important case of a uniform lying FLC helix in the slab geometry with the smectic layers normal to the substrates and

(3)

where is the electric field applied across the cell, is illustrated in Fig. 1. This is the geometry of surface stabilized FLCs (SSFLCs) pioneered by Clark and Lagerwall in Clark and Lagerwall (1980). They studied electro-optic response of FLC cells confined between two parallel plates subject to homogeneous boundary conditions and made thin enough to suppress the bulk FLC helix.

It was found that such cells exhibit high-speed, bistable electro-optical switching between orientational states stabilized by surface interactions. The response of FLCs to an applied electric field is characterized by fast switching times due to linear coupling between the field and the spontaneous ferroelectric polarization

(4)

where is the polarization unit vector. There is also a threshold voltage necessary for switching to occur and the process of bistable switching is typically accompanied by a hysteresis.

Figure b also describes the geometry of deformed helix FLCs (DHFLCs) as it was introduced in Beresnev et al. (1989). This case will be of our primary concern.

In DHFLC cells, the FLC helix is characterized by a short submicron helix pitch,  m, and a relatively large tilt angle,  deg. By contrast to SSFLC cells, where the surface induced unwinding of the bulk helix requires the helix pitch of a FLC mixture to be greater than the cell thickness, a DHFLC helix pitch is 5-10 times smaller than the thickness. This allows the helix to be retained within the cell boundaries.

Electro-optical response of DHFLC cells exhibits a number of peculiarities that make them useful for LC devices such as high speed spatial light modulators Abdulhalim and Moddel (1991); Cohen et al. (1997); Pozhidaev et al. (2000, 2013, 2014), colour-sequential liquid crystal display cells Hedge et al. (2008) and optic fiber sensors Brodzeli et al. (2012). The effects caused by electric-field-induced distortions of the helical structure underline the mode of operation of such cells. In a typical experimental setup, these effects are probed by performing measurements of the transmittance of normally incident linearly polarized light through a cell placed between crossed polarizers.

A more general case of oblique incidence has not received a fair amount of attention. Theoretically, a powerful tool to deal with this case is the transfer matrix method which has been widely used in studies of both quantum mechanical and optical wave fields Markoš and Soukoulis (2008); Yariv and Yeh (2007). In this work we apply the method for systematic treatment of the technologically important case of DHFLCs with subwavelength pitch also known as the short-pitch DHFLCs.

Recently, the transfer matrix approach to polarization gratings was employed to define the effective dielectric tensor of short-pitch DHFLCs Kiselev et al. (2011) that gives the principal values and orientation of the optical axes as a function of the applied electric field. Biaxial anisotropy and rotation of the in-plane optical axes produced by the electric field can be interpreted as the orientational Kerr effect Pozhidaev et al. (2013, 2014).

It can be expected that the electric field dependence of the effective dielectric tensor will also manifest itself as electric-field-induced transformations of the polarization-resolved angular (conoscopic) patterns in the observation plane after the DHFLC cells illuminated by convergent light beam. These patterns are represented by the two-dimensional (2D) fields of polarization ellipses describing the polarization structure behind the conoscopic images Kiselev (2007a); Kiselev et al. (2008).

As it was originally recognized by Nye Nye (1983); Nye and Hajnal (1987); Nye (1999), the key elements characterizing geometry of such Stokes parameter fields are the polarization singularities that play the fundamentally important role of structurally stable topological defects (a recent review can be found in Ref. Dennis et al. (2009)). In particular, the polarization singularities such as the C points (the points where the light wave is circular polarized) and the L lines (the curves along which the polarization is linear) frequently emerge as the characteristic feature of certain polarization state distributions. For nematic and cholescteric (chiral nematic) liquid crystals, the singularity structure of the polarization-resolved angular patterns is generally found to be sensitive to both the director configuration and the polarization characteristics of incident light Kiselev (2007a); Kiselev et al. (2008); Egorov and Kiselev (2010).

In this study, we consider the polarization-resolved angular patterns of DHFLC cells as the Stokes parameter fields giving detailed information on the incidence angles dependence of the polarization state of light transmitted through the cells. In particular, we explore how the polarization singularities transform under the action of the electric field. Our analysis will utilize the transfer matrix approach in combination with the results for the effective dielectric tensor of biaxial FLCs evaluated using an improved technique of averaging over distorted helical structures. We also emphasize the role of phase singularities of a different kind and discuss the electro-optic behavior of DHFLCs near the exceptional point where the condition of zero-field isotropy is fulfilled.

The layout of the paper is as follows. In Sec. II we introduce our notations and describe the transfer matrix formalism rendered into the matrix form suitable for our purposes. This formalism is employed to deduce a number of the unitarity and symmetry relations with emphasis on the planar anisotropic structures that represent DHFLC cells and posses two optical axes lying in the plane of substrates. In Sec. III we evaluate the effective dielectric tensor of DHFLC cells, discuss the orientational Kerr effect and show that electro-optic response of DHFLC cells is enhanced near the exceptional point determined by the condition of zero-field isotropy. Geometry of the polarization-resolved angular patterns emerging after DHFLC cells is considered in Sec. IV. After providing necessary details on our computational approach and the polarization singularities, we present the numerical results describing how the singularity structure of polarization ellipse fields transforms under the action of the electric field. Finally, in Sec. V we draw the results together and make some concluding remarks. Details on some technical results are relegated to Appendixes AC.

Figure 2: Four-wave geometry with two incoming (incident) waves, and , impinging onto the entrance () and exit () faces, respectively.

Ii Transfer matrix method and symmetries

In order to describe both the electro-optical properties and the polarization-resolved angular patterns of deformed helix ferroelectric liquid crystal layers with subwavelength pitch we adapt a general theoretical approach which can be regarded as a modified version of the well-known transfer matrix method Markoš and Soukoulis (2008); Yariv and Yeh (2007) and was previously applied to study the polarization-resolved conoscopic patterns of nematic liquid crystal cells Kiselev (2007a); Kiselev et al. (2008); Kiselev and Vovk (2010). This approach has also been extended to the case of polarization gratings and used to deduce the general expression for the effective dielectric tensor of DHFLC cells Kiselev et al. (2011).

In this section, we present the transfer matrix approach as the starting point of our theoretical considerations, with emphasis on its general structure and the symmetry relations. The analytical results for uniformly anisotropic planar structures representing homogenized DHFLC cells are given in Appendix B.

We deal with a harmonic electromagnetic field characterized by the free-space wave number , where is the frequency (time-dependent factor is ), and consider the slab geometry shown in Fig. 2. In this geometry, an optically anisotropic layer of thickness is sandwiched between the bounding surfaces (substrates): and (the axis is normal to the substrates) and is characterized by the dielectric tensor and the magnetic permittivity

Further, we restrict ourselves to the case of stratified media and assume that the electromagnetic fields can be taken in the following factorized form

(5)

where the vector

(6)

represents the lateral component of the wave vector. Then we write down the representation for the electric and magnetic fields, and ,

(7)

where the components directed along the normal to the bounding surface (the axis) are separated from the tangential (lateral) ones. In this representation, the vectors and are parallel to the substrates and give the lateral components of the electromagnetic field.

Substituting the relations (7) into the Maxwell equations and eliminating the components of the electric and magnetic fields gives equations for the tangential components of the electromagnetic field that can be written in the following matrix form Kiselev et al. (2008, 2011):

(8)

where is the differential propagation matrix and its block matrices are given by

(9a)
(9b)
(9c)

General solution of the system (8)

(10)

can be conveniently expressed in terms of the evolution operator which is also known as the propagator and is defined as the matrix solution of the initial value problem

(11a)
(11b)

where is the identity matrix. Basic properties of the evolution operator are reviewed in Appendix A.

ii.1 Input-output relations

In the ambient medium with and , the general solution (10) can be expressed in terms of plane waves propagating along the wave vectors with the tangential component (6). For such waves, the result is given by

(12)
(13)

where is the eigenvector matrix for the ambient medium given by

(14)
(15)
(16)

are the Pauli matrices

(17)

From Eq. (12), the vector amplitudes and correspond to the forward and backward eigenwaves with and , respectively. Figure 2 shows that, in the half space before the entrance face of the layer , these eigenwaves describe the incoming and outgoing waves

(18)

whereas, in the half space after the exit face of the layer, these waves are given by

(19)

In this geometry, there are two plane waves, and , incident on the bounding surfaces of the anisotropic layer, and , respectively. Then the standard linear input-output relations

(20)

linking the vector amplitudes of transmitted and reflected waves, and with the amplitude of the incident wave, through the transmission and reflection matrices, and , assume the following generalized form:

(21)

where is the matrix —  the so-called scattering matrix — that relates the outgoing and incoming waves; () is the transmission (reflection) matrix for the case when the incident wave is incoming from the half space bounded by the entrance face, whereas the mirror symmetric case where the incident wave is impinging onto the exit face of the sample is described by the transmission (reflection) matrix (). So, we have

(22a)
(22b)
(22c)

It is our task now to relate these matrices and the evolution operator given by Eq. (11). To this end, we use the boundary conditions requiring the tangential components of the electric and magnetic fields to be continuous at the boundary surfaces: and , and apply the relation (11) to the anisotropic layer of the thickness to yield the following result

(23)

ii.2 Transfer matrix

On substituting Eqs. (12) into Eq. (23) we have

(24)

where the matrix linking the electric field vector amplitudes of the waves in the half spaces and bounded by the faces of the layer will be referred to as the transfer (linking) matrix. The expression for the transfer matrix is as follows

(25)

where is the rotated operator of evolution. This operator is the solution of the initial value problem (11) with replaced with .

From Eqs. (21) and (24), the block structure of the transfer matrix can be expressed in terms of the transmission and reflection matrices as follows

(26)

Similarly, for inverse of the transfer matrix,

(27)

we have

(28)

ii.3 Symmetries

In Appendix A, it is shown that, for non-absorbing media with symmetric dielectric tensor, , the operator of evolution satisfies the unitarity relation (95). By using Eq. (95) in combination with the algebraic identity

(29a)
(29b)

where , for the eigenvector matrix given in Eq. (14), we can deduce the unitarity relation for the transfer matrix (25)

(30)

The unitarity relation (30) for non-absorbing layers can now be used to derive the energy conservation laws

(31a)
(31b)

where a dagger and the superscript will denote Hermitian conjugation and matrix transposition, respectively, along with the relations for the block matrices

(32a)
(32b)
(32c)

Note that Eqs. (32b) and (32c) can be conveniently rewritten in the following form

(33a)
(33b)

so that multiplying these identities and using the energy conservation law (31a) gives the relations (31b).

In the translation invariant case of uniform anisotropy, the matrix is independent of and the operator of evolution is given by

(34)

Then, the unitarity condition Kiselev et al. (2008)

(35)

can be combined with Eq. (30) to yield the additional symmetry relations for

(36)

where an asterisk will indicate complex conjugation, that give the following algebraic identities for the transmission and reflection matrices:

(37)
(38)

It can be readily seen that the relation for the transposed matrices (31b) can be derived by substituting Eq. (37) into the conservation law (31a).

For the important special case of uniformly anisotropic planar structures with , the algebraic structure of the transfer matrix is described in Appendix B. Equation (107) shows that the symmetry relations (36) remain valid even if the dielectric constants are complex-valued and the medium is absorbing. Since identities (37) are derived from Eqs. (36) and (II.2) without recourse to the unitarity relations, they also hold for lossy materials.

Similar remark applies to the expression for inverse of the transfer matrix (109). From Eq. (109), it follows that the relation between the transmission (reflection) matrix, (), and its mirror symmetric counterpart () can be further simplified and is given by

(39)

From Eqs. (39) and (37), we have the relation for the transposed reflection matrices

(40)

whereas the transmission matrices are symmetric.

Iii Electro-optics of homogenized DHFLC cells

We now pass on to the electro-optical properties of DHFLC cells and extend the results of Ref. Kiselev et al. (2011) to the case of biaxial ferroelectric liquid crystals with subwavelength pitch. In addition, the theoretical treatment will be significantly improved by using an alternative fully consistent procedure to perform averaging over distorted FLC helix that goes around the limitations of the first-order approximation.

iii.1 Effective dielectric tensor

We consider a FLC film of thickness with the axis which, as is indicated in Fig. 1, is normal to the bounding surfaces: and , and introduce the effective dielectric tensor, , describing a homogenized DHFLC helical structure. For a biaxial FLC, the components of the dielectric tensor, , are given by

(41)

where , is the Kronecker symbol; () is the th component of the FLC director (unit polarization vector) given by Eq. (1) (Eq. (4)); are the anisotropy parameters and () is the anisotropy (biaxiality) ratio. Note that, in the case of uniaxial anisotropy with , the principal values of the dielectric tensor are: and , where () is the ordinary (extraordinary) refractive index and the magnetic tensor of FLC is assumed to be isotropic with the magnetic permittivity . As in Sec. II (see Fig. 2), the medium surrounding the layer is optically isotropic and is characterized by the dielectric constant , the magnetic permittivity and the refractive index .

At , the ideal FLC helix

(42)

where is the free twist wave number and is the equilibrium helical pitch, is defined through the azimuthal angle around the smectic cone (see Fig. 1 and Eq. (3)) and represents the undistorted structure. For sufficiently small electric fields, the standard perturbative technique applied to the Euler-Lagrange equation gives the first-order expression Chigrinov (1999); Hedge et al. (2008) for the azimuthal angle of a weakly distorted helical structure

(43)

where is the electric field parameter linearly proportional to the ratio of the applied and critical electric fields: , and .

According to Ref. Kiselev et al. (2011), normally incident light feels effective in-plane anisotropy described by the averaged tensor, :

(44)
(45)

where , and the effective dielectric tensor

(46)

can be expressed in terms of the averages

(47)
(48)

as follows

(49)

General formulas (44)-(49) give the zero-order approximation for homogeneous models describing the optical properties of short pitch DHFLCs Kiselev et al. (2011); Pozhidaev et al. (2013). Assuming that the pitch-to-wavelength ratio is sufficiently small, these formulas can now be used to derive the effective dielectric tensor of homogenized short-pitch DHFLC cell for both vertically and planar aligned FLC helix. The results for vertically aligned DHFLC cells were recently published in Ref. Pozhidaev et al. (2013) and we concentrate on the geometry of planar aligned DHFLC helix shown in Fig. 1. For this geometry, the parameters needed to compute the averages (see Eq. (44)), (see Eq. (47)) and (see Eq. (48)) are given by

(50)
(51)
(52)

Formulas (53) can now be inserted into Eqs. (49) to yield the explicit expressions for the elements of the dielectric tensor (46):

(53a)
(53b)
(53c)
(53d)

iii.2 Orientational Kerr effect

The simplest averaging procedure previously used in Refs. Abdulhalim and Moddel (1991); Kiselev et al. (2011); Pozhidaev et al. (2013) involves substituting the formula for a weakly distorted FLC helix (43) into Eqs. (53) and performing integrals over . This procedure thus heavily relies on the first-order approximation where the director distortions are described by the term linearly proportional to the electric field (the second term on the right hand side of Eq. (43)). Quantitatively, the difficulty with this approach is that the linear approximation may not be suffice for accurate computing of the second-order contributions to the diagonal elements of the dielectric tensor (53). In this approximation, the second-order corrections describing the helix distortions that involve the change of the helix pitch have been neglected.

In order to circumvent the problem, in this paper, we apply an alternative approach that allows to go beyond the first-order approximation without recourse to explicit formulas for the azimuthal angle. This method is detailed in Appendix C. The analytical results (137) substituted into Eqs. (53) give the effective dielectric tensor in the following form:

(54)

The zero-field dielectric constants, and , that enter the tensor (54) are defined in Eqs. (138) and (139), respectively, and can be conveniently rewritten as follows

(55a)
(55b)

A similar result for the coupling coefficients , and (see Eq. (140)) reads

(56a)
(56b)
(56c)

Note that, following Ref. Pozhidaev et al. (2013), we have used the relation (136) to introduce the electric field parameter

(57)

where is the dielectric susceptibility of the Goldstone mode Carlsson et al. (1990); Urbanc et al. (1991).

The above dielectric tensor is characterized by the three generally different principal values (eigenvalues) and the corresponding optical axes (eigenvectors) as follows

(58)
(59)
(60)

where

(61)
(62)

The in-plane optical axes are given by

(63)

From Eq. (54), it is clear that, similar to the case of uniaxial FLCs studied in Ref. Kiselev et al. (2011), the zero-field dielectric tensor is uniaxially anisotropic with the optical axis directed along the twisting axis . The applied electric field changes the principal values (see Eqs. (59) and (60)) so that the electric-field-induced anisotropy is generally biaxial. In addition, the in-plane principal optical axes are rotated about the vector of electric field, , by the angle given in Eq. (63).

In the low electric field region, the electrically induced part of the principal values is typically dominated by the Kerr-like nonlinear terms proportional to , whereas the electric field dependence of the angle is approximately linear: . This effect is caused by the electrically induced distortions of the helical structure and bears some resemblance to the electro-optic Kerr effect. Following Refs. Pozhidaev et al. (2013, 2014), it will be referred to as the orientational Kerr effect.

It should be emphasized that this effect differs from the well-known Kerr effect which is a quadratic electro-optic effect related to the electrically induced birefringence in optically isotropic (and transparent) materials and which is mainly caused by the electric-field-induced orientation of polar molecules Weinberger (2008). By contrast, in our case, similar to polymer stabilized blue phase liquid crystals Yan et al. (2010, 2013), we deal with the effective dielectric tensor of a nanostructured chiral smectic liquid crystal. This tensor (53) is defined through averaging over the FLC orientational structure.

Figure 3: (Color online) Transmittance of light passing through crossed polarizers, , as a function of the applied electric field at the wavelength  nm for the DHFLC cell of thickness  m filled with the FLC mixture FLC-576A Kiselev et al. (2011). Parameters of the mixture are: () is the ordinary (extraordinary) refractive index and  deg is the tilt angle. The experimental points are marked by squares. The results obtained using the linear approximation are shown as dashed line Dashed and solid lines represent the theoretical curves computed using the linear approximation Kiselev et al. (2011) and the improved method of averaging with  V/m, respectively.

Typically, in experiments dealing with the electro-optic response of DHFLC cells, the transmittance of normally incident light passing through crossed polarizers is measured as a function of the applied electric field. For normal incidence, the transmission and reflection matrices can be easily obtained from the results given in Appendix B by substituting Eq. (121) into Eqs. (107)- (108). When the incident wave is linearly polarized along the axis (the helix axis), the transmittance coefficient

(64)
(65)

where is the thickness parameter, describes the intensity of the light passing through crossed polarizers. Note that, under certain conditions such as , and the transmittance (64) can be approximated by simpler formula

(66)

where is the difference in optical path of the ordinary and extraordinary waves known as the phase retardation.

In Ref. Kiselev et al. (2011), the relation (64) was used to fit the experimental data using the theory based on the linear approximation for the helix distortions (see Eq. (43)). These results are reproduced in Figure 3 along with the theoretical curve computed using the modified averaging technique. From Fig. 3, it is seen that, in the range of relatively high voltages, the averaging method described in Appendix C improves agreement between the theory and the experiment, whereas, at small voltages, the difference between the fitting curves is negligibly small.

(a)
(b)
Figure 4: (Color online) Coupling coefficients as a function of the smectic tilt angle at for (a) a uniaxially anisotropic FLC with and (b) a biaxially anisotropic FLC with .

iii.3 Effects of smectic tilt angle

Given the anisotropy and biaxiality ratios, and , the zero-field dielectric constants  (55) and the coupling coefficients (56) are determined by the smectic tilt angle, . Figure 4 shows how the coupling coefficients depend on for both uniaxially and biaxially anisotropic FLCs.

As it can be seen in Fig. a, in the case of conventional FLCs with , all the coefficients are positive and the difference of the coupling constants that define the electrically induced part of (see Eq. (62)) is negative at .

From Eqs. (59) and (60), it follows that, at , the principal values of dielectric constants and are decreasing functions of the electric field parameter so that anisotropy of the effective dielectric tensor (54) is weakly biaxial. In addition, for non-negative and , the azimuthal angle of in-plane optical axis, , given in Eq. (63) increases with from zero to .

Figure b demonstrates that this is no longer the case for biaxial FLCs. It is seen that, at , the coupling coefficient and the difference both change in sign when the tilt angle is sufficiently small. At such angles, the dielectric constant increases with and the electric field induced anisotropy of DHFLC cell becomes strongly biaxial. When and are positive, electric field dependence of the azimuthal angle is non-monotonic and the angle decays to zero in the range of high voltages where .

(a)
(b)
Figure 5: (Color online) Zero-field dielectric constants as a function of the smectic tilt angle at for (a) a uniaxially anisotropic FLC with and (b) a biaxially anisotropic FLC with .

iii.4 Zero-field isotropy and electro-optic response near exceptional point

At , the zero-field anisotropy is uniaxial and is described by the dielectric constants, and , given in Eq. (55). In Fig. 5, these constants are plotted against the tilt angle. It is shown that, at small tilt angles, the anisotropy is positive. It decreases with and the zero-field state becomes isotropic when, at certain critical angle , the condition of zero-field isotropy

(67)

is fulfilled and . So, the angle can be referred to as the isotropization angle. In what follows we discuss peculiarities of the electro-optic response in the vicinity of the isotropization point where is proportional to (see Eq. (62)) and the Kerr-like regime breaks down.

Mathematically, the isotropization point represents a square root branch-point singularity of the eigenvalues (60) of the dielectric tensor which is known as the exceptional point Kato (1995); Heiss and Sanino (1990); Heiss (2000). In the electric field dependence of the in-plane dielectric constants, and , this singularity reveals itself as a cusp where the derivatives of with respect to are discontinuous. More precisely, we have

(68)

As is illustrated in Fig. a, the cusp is related to the effect of reconnection of different branches representing solutions of an algebraic equation.

(a)  deg
(b)  deg
(c)  deg
Figure 6: (Color online) Principal values of the effective dielectric tensor as a function of the electric field parameter at and for different values of the smectic tilt angles.

Since the azimuthal angle is undetermined at

(69)

the isotropization point also represents a phase singularity. The electric field dependence of is thus discontinuous and the relation

(70)

describes its jumplike behaviour at . This behaviour is demonstrated in Fig. 7.

(a)
(b)
Figure 7: (Color online) Principal axis azimuthal angle versus the electric field parameter at for (a) a uniaxially anisotropic FLC with and (b) a biaxially anisotropic FLC with .

We can now use Eq. (55) and write down the condition of zero-field isotropy (67) in the following explicit form:

(71)

The case of a uniaxially anisotropic FLC with can be treated analytically. In this case, it is not difficult to check that gives the special solution of Eq. (71) that does not depend on the tilt angle and corresponds to an isotropic material with . Another solution is given by the relation