Electromagnetic Waves in Crystals: The Presence of Exceptional Points

Although the investigation of the propagation of electromagnetic waves in crystals dates back to the 19th century, the presence of singular optic axes in optically anisotropic materials has not been fully explored until now. Along such an axis, either a left or a right circular polarized wave can propagate without changing its polarization state. More generally, these singular optic axes belong to exceptional points (EPs) in the momentum space and correspond to a simultaneous degeneration of the eigenmodes and their propagation properties. Herein, a comprehensive discussion on EPs in optically anisotropic materials, their occurrence, and properties as well as the properties of the electromagnetic waves propagating along such EPs is presented. The presence of such EPs, their spatial and spectral distribution in bulk, and semi‐infinite and finite crystals are discussed. It is shown that the presence of interfaces has a strong impact on the presence of the EPs and their spatial distribution. At an EP, the propagation of an arbitrarily polarized wave cannot be described by a superposition of two eigenmodes, as typically described in textbooks. This leads to singularities if the reflection and transmission coefficients have to be calculated. Here, two approaches are presented to overcome these limitations.

singular optic axes can coincide with each other so that monoclinic and triclinic crystals can have three, two, and one distinguished axes. [11,12]The spectral dispersion of the singular optic axes for real materials has been discussed recently for β-Ga 2 O 3 , which has a monoclinic crystal structure, and the formation of triaxial points has been experimentally confirmed. [8]t was also shown that optically biaxial materials, namely, the orthorhombic, monoclinic, and triclinic crystal structure, can be distinguished in the absorbing spectral range by the number of different complex refractive indices of their, generally separate, four singular optic axes. [8,13]n general, these singular optic axes correspond to exceptional points (EPs) in the momentum space.The presence of such EPs is not limited to bulk crystals only and they appear in different optical systems, [14,15] such as 1D confined microresonators, [16][17][18][19] photonic crystals, [20] and ring resonators. [21,22]The dynamics of wave packets at an EP has recently been studied theoretically by D. D. Solnyshkov et al. [23] Mathematically, the reason for the existence of these EPs is the non-Hermitian nature of the underlying physics, [24,25] e.g., in the case of bulk crystal of the non-Hermitian character of the wave equation.Especially in the last years, the study of the physics of EPs has become a focus of research. [14,15]From the application point of view, EPs are particularly interesting for sensing applications because the response is quite sensitive to changes in the vicinity of an EP. [26]n this work, we discuss in detail EPs in optically anisotropic materials, their occurrence, spatial distribution, and the properties of the corresponding waves.We show that the crystal symmetry as well as the presence of interfaces have a strong impact.While in infinitely thick bulk single crystals EPs appear only for optically biaxial materials, namely those having an orthorhombic, monoclinic, or triclinic crystal structure, the situation changes if an interface is present.In this case, we show that EPs can exist also in optically uniaxial materials and even in the transparent spectral range, whereas in bulk crystals the presence of EPs is limited to the absorption range.As a further consequence of the interface, instead of eight well-defined EPs, trajectories of EPs can appear.Interestingly, while the properties of the EPs in bulk crystals are intrinsically determined by the dielectric function, we will show that these properties can be tuned if interfaces are present.This is of special interest for applications making use of EPs.
Another aspect, which is discussed here in detail, is the propagation of an arbitrarily polarized wave at an EP.Typically, an arbitrarily polarized wave is described by a superposition of the corresponding eigenmodes.However, due to the degeneration of the eigenmodes at the EP, this approach cannot be applied here.As a consequence, also the reflection and transmission of an arbitrarily polarized wave at an interface cannot be calculated by means of these eigenmodes if an EP is involved.Here, we present two generalized approaches in order to describe the propagation of an arbitrarily polarized wave at an EP.By means of these approaches, the reflection and transmission coefficients at an interface can be calculated even in the case when an EP is involved.
The article is organized as follows: In Section 2, we discuss the wave properties and EPs bulk crystals, while in Section 3 and 4 the impact of the interfaces on the EP is discussed.In Section 5 we present two formalisms, which describe the propagation of an arbitrarily polarized electromagnetic wave within a crystal at an EP.This formalism is then applied in Section 6 to present a general approach, which allow the calculation of the reflection and transmission properties of an electromagnetic wave at interfaces without singularities when EPs are involved.In Section 7, peculiarities between bulk, semi-infinite, and finite crystals are discussed.The results are then summarized in Section 8.

Wave Equation
For optically nonactive and nonmagnetic materials, the wave equation is given by A solution of this equation can be obtained by using plane waves for the electric field E and the displacement D, i.e., E, D ∝ e i kxÀωt ð Þ with the angular frequency ω and the wave vector k.Typically, the phase front and the amplitude front in bulk crystals are coplanar, i.e., the real and imaginary parts of the wave vector k point in the same direction.These waves are named as homogeneous or uniform plane waves. [27]In this case, it is convenient to define a coordinate system such that the propagation direction is along the z 0 -direction.The wave equation then becomes 1 0 0 0 1 0 0 0 0 with ñ ¼ n þ iκ being the complex refractive index, where n and κ represent the refractive index and the extinction coefficient, respectively.Note that the prime in Equation ( 2), as well as in the following equations, denotes that these quantities are defined in the framework with the z 0 -direction along the propagation direction.The transformation between the two coordinate systems, e.g., between the laboratory (x-y-z-system) and the one defined by the wave propagation along the z 0 -direction (x 0 -y 0 -z 0 -system), is done by means of a rotation matrix where θ and ϕ represent the rotation angle around the y-and the subsequent z-axis, respectively.In doing so, the dielectric tensor as well as the electric field and displacement transform to and the propagation direction in the x-y-z-system is given by r ¼ sin θ cos ϕ sin θ sin ϕ cos θ In general, Equation 2 represents a coupled system of three equations.However, not all three equations are independent of each other.In the case of the electric field, only two of the three spatial components are independent of each other.In the system defined by z 0 , it is convenient to choose E 0 x 0 and E 0 y 0 as independent quantities, so that E 0 z 0 is given by In this case, Equation (2) reduces to a system of two coupled equations, i.e.
with Ê0 ¼ E 0 x 0 , E 0 y 0 T and the "reduced" dielectric tensor Thus, finding the solution of the wave equation is equivalent to solving an eigenvalue problem.
The subscript 2 Â 2 denotes the block matrix consisting of the first two rows and columns of ε 0À1 , i.e., ε 0À1 2Â2 ½ ij ¼ ε 0À1 ½ ij with i, j ¼ x 0 , y 0 .The solution of Equation (7) and (9) leads to a quadratic equation in ñ2 , and the difference of the two solution is given by In the transparent spectral region, the components of the dielectric tensor are real-valued and thus for an optic axis with Δñ ¼ 0, each of the quadratic terms in the square root must be zero.In the absorption spectral range, the dielectric function is complex-valued and thus for a degeneracy of the complex refractive index has to be fulfilled.
It can be easily seen that Equation (11) is fulfilled for isotropic materials for all directions and for optically uniaxial materials for propagation along the symmetry axis, i.e., θ ¼ 0 and ε 0 ij ¼ 0. For these crystal symmetries, the degeneracy of the (complex) refractive index leads to two orthogonally polarized eigenstates, in the transparent as well as in the absorption spectral range.These directions in which the (complex) refractive index is degenerate are called optic axes. [1]Thus, in the entire spectral range, the propagation properties of the electromagnetic wave along an optic axis are independent of the polarization.
For optically uniaxial materials, the reduced dielectric tensor (Equation ( 8)) has a diagonal shape for all propagation directions and thus, even if the eigenvalue, i.e., the complex refractive index is degenerate, the eigenstates are not degenerate and an orthogonal basis can be chosen (cf., Section A).For optically biaxial crystals, i.e., those belonging to the orthorhombic, monoclinic, or triclinic crystal structure, the situation is different.In this case, the "reduced" dielectric tensor ε0 has in general nonvanishing off-diagonal elements.In the transparent spectral region, the dielectric tensor is symmetric, and thus it can be diagonalized and its eigenvalues, i.e., the refractive index, can degenerate.However, the corresponding eigenvectors, i.e., the electric field vectors, are still orthogonal to each other (cf., Section A).It is well known that in this spectral region, four directions exist, where in each case two of them are pointing opposite directions and thus forming an axis.This axis is called the optic axis and these materials are called optically biaxial.In the absorption spectral range, the dielectric tensor cannot be diagonalized by means of realvalued rotation matrices.Thus, a degeneration of the complex refractive index (eigenvalue) is accompanied by a simultaneous degeneration of the electric field (eigenvector), cf., (Appendix A).Note, the degeneration of the eigenvalues and eigenvectors are based on algebraic consideration and thus the solution of Equation (11) does not require any assumption on the tensor elements of the dielectric function and can be fulfilled exactly.Any further requirements on the dielectric function, e.g., the fulfillment of the Kramers-Kronig relation or sum rules, only have an impact on the orientation where the degeneration takes place but not on the presence of the degeneration itself.Recently, Grundmann et al. [28] gave an analytical expression for the solution of Equation (11), i.e., the orientations where the eigenvalues and eigenvectors are degenerate.
If we neglect the presence of a magnetic field or optical activity, the dielectric tensor is symmetric and the degenerate electric fields are either left or right circularly polarized (cf., Section A). [7,28] In this case, the solutions of Equation ( 13) can be expressed by the solution of two quartic polynomials, in each case one representing the left and right circular polarized eigenmode.The corresponding polynomials are given by [28] S AE ϕ, θ ð Þ ¼ X 4 j¼0 l j x j κ ∓j ¼ 0 (12)   with and x ¼ e iϕ and κ ¼ 1= i tan θ=2 ð Þ .The sign of the index defines the corresponding polynomial for the right (þ) or left (À) circular polarized eigenmode.An analytical expression for the roots can be found in ref. [28] Note, in the case that the dielectric tensor is not symmetric anymore, e.g., due to the presence of an external magnetic field, the eigenstates for the propagation along the singular optic axis are elliptically and no longer circularly polarized. [28]Then, the directions of the singular optic axis cannot be described by a quartic polynomial as given by Equation (12).The corresponding equation can be found in ref. [28].
As can be seen from Equation (12), the solution for the direction of the right circular polarized eigenmode corresponds to the opposite directions for the left circular polarized eigenmode, and vice versa, and so one pair of directions for the left and circular polarized waves forms an axis.In general, there are four singular optic axes.However, depending on the dielectric function, these axes can coincide with each other, reducing the number of distinct axes.A detailed discussion of the number of (singular) optical axes and the condition for the dielectric tensor for monoclinic and triclinic material symmetry was done by Golovina et al. [11,12] In the following, we express the polarization of the wave by the Stokes vector S ¼ S 0 , S 1 , S 2 , S 3 ð Þ T .The components S i are defined by [29] S where S 0 represents the entire intensity of the wave, and the components S 1 , S 2 , and S 3 represent the difference between x-and y-(I x À I y ), þ45 ∘ and À45 ∘ (I þ45 ∘ À I À45 ∘ ) as well as right and left circular (I R À I L ) polarized intensities of the wave.

Symmetry Properties
The solution of the wave equation for homogeneous plane waves (Equation ( 7)) generally yields two solutions in ñ2 and the wave vector of the corresponding wave is given by k ¼ AE2π ñr=λ, with r the unit vector representing the propagation direction.The þ (À) sign represents the forward (backward) propagating wave.Thus, the degeneration of the refractive index occurs for the forward and backward traveling waves simultaneously and therefore the positive and negative propagation direction represents an axis, the so-called singular optic axis.The corresponding eigenstates of the forward and backward propagating waves have the opposite polarization, which is expressed by the opposite sign of the Stokes vector component S 3 .Note, the Stokes vector components S 1 and S 2 , which represent the linear polarization, are zero for the eigenstate propagating along a singular optic axis.Thus, if (ϕ 0 , θ 0 ) represents a direction of a singular optic axis, then the following relations hold between the direction vector (r), the complex refractive index (ñ), and the circular polarization (S 3 ) of the eigenmode between the singular optic axis The symmetry properties given by Equation ( 15) are independent of the crystal symmetry.Exemplarily for the three crystal symmetries, which are optically biaxial and thus exhibiting singular optic axes, the orientation of these axes is calculated from experimentally determined dielectric functions for orthorhombic KTiOPO 4 (KTP), monoclinic Ga 2 O 3 (β-phase), and triclinic K 2 Cr 2 O 7 .The dielectric functions of these materials were determined by means of spectroscopic ellipsometry and can be found for KTP, β-Ga 2 O 3 , and K 2 Cr 2 O 7 in ref. [30-32], respectively.Note that in the following we use a different coordinate system for β-Ga 2 O 3 as in ref. [31].Here, we choose the coordinate system in such way that the symmetry axis coincides with the z-axis, i.e., êy ka, êz kb, and êx ¼ êy Â êz , with êi representing the unit vector in the i th direction.
For a selected energy, the orientations of the singular optic axes for KTP, β-Ga 2 O 3 , and K 2 Cr 2 O 7 are given in Figure 1, where the difference of the complex refractive index of the two eigenmodes and the degree of the circular polarization (S 3 ) are shown.The symmetry relations mentioned above (Equation (15b) and (15c)) are nicely reproduced.Furthermore, it can be easily seen that a higher crystal symmetry can cause further symmetry relations, which will be discussed now.

Orthorhombic Crystal Structure
In the case of the orthorhombic crystal structure, the crystallographic axes are perpendicular to each other and thus there exists a well-defined Cartesian coordinate system.The tensor of the dielectric function in this case is a diagonal matrix, i.e., ε ij ¼ 0 for i 6 ¼ j.The enhanced symmetry compared to the monoclinic and triclinic crystal structure is also reflected in the properties of the singular optic axes.For this crystal structure, all axes lie on the surface of a cone.If ϕ 0 , θ 0 ð Þare the azimuth and polar angle of one direction of a singular optic axis, then the corresponding angles of the other three directions in this hemisphere are and the corresponding complex refractive index and the circular polarization of the eigenmodes are Note, the first index in Equation (17b) denotes the component of the Stokes vector, whereas the second one denotes the number of the singular optic axis.
As can be seen from Equation ( 16) and (17), in orthorhombic crystals all four singular optic axes are related to each other, i.e., all have the same refractive index and the orientation and polarization of one axis determines the orientation of the other three.Besides the presence of the four singular optic axes, a reduced number of distinguished axes can also appear, as was also pointed out by Golovina et al. [11] In orthorhombic materials, biaxial (B) and uniaxial (U ) points can be present.
• B 2 : two classical optical axes The singular optic axes are in a plane defined by two dielectric (coordinate) axes, i.e., -θ ∈ 0, π=2, π f gand ϕ ¼ 0 (x-z-plane), or -θ ∈ 0, π=2, π f gand ϕ ¼ π=2 (y-z-plane), or -θ ¼ π=2 and ϕ ∈ 0, π=2 f g(x-y-plane) Here, two singular optic axes, one with a left and the other one with a right circular polarized eigenmode, coincide with each other and form a classical optic axis.This situation appears if the ratio of the major axes of the ellipsoid defined by the real and imaginary part of the dielectric tensor is equal, i.e., Reε xx ∶Reε yy ∶Reε zz ¼ Imε xx ∶Imε yy ∶Imε zz .Note, this is then equivalent to the well-known condition that the classical optic axes in an optically biaxial material in the transparent spectral range are in the plane defined by dielectric axes with the largest and smallest dielectric constant. [1] U 2 : one classical optical axis The optic axes are pointing in the x, y, or z direction, i.e., -θ ¼ π=2 and ϕ ¼ 0 (x-direction), or -θ ¼ π=2 and ϕ ¼ π=2 (y-direction), or -θ ∈ 0, π f g (z-direction) As all axes then point in the same direction, this leads to a superposition of optical axes with a left and right circular polarization, and a classical optic axis is formed.In this case, the material is optically uniaxial at a particular energy.
Exemplarily for this crystal structure, the polar (θ) and azimuth angle (ϕ) of the (singular) optic axes as a function of the energy are shown for KTiOPO 4 (KTP) in Figure 2a.As soon as the absorption sets in, the two optic axes split up and four singular optic axes are present.The orientation and the polarization of the eigenmode follow the symmetry relations given by Equation ( 16) and (17).These symmetry relations are also reflected by the twofold rotation symmetry of the orientation distribution and, if the polarization of the eigenmode is neglected, the presence of three mirror planes.This can be nicely seen in the stereographic projection of the distribution of the axes (Figure 2b).Note, the twofold rotation symmetry exists for all three dielectric axes and all planes defined by the dielectric axes represent a mirror plane if the polarization of the eigenmode is neglected.Thus, in an orthorhombic crystal structure, three C 2 rotation axes and three mirror planes exist regarding the spatial distribution of the singular optic axes.In the case of KTP, in the investigated spectral range, four distinct energies exist, namely, at E ≈ 5.85 eV, 6.60 eV, 7.42 eV, and 8.22 eV, which are labeled as i À iv and are indicated by the vertical dashed lines in Figure 2a.At these energies, the singular optic axes coincide pairwise with each other and, as a consequence of the symmetry considerations mentioned above, a biaxial point B 2 is formed, i.e., two classical optic axes with two orthogonal eigenstate exist along this direction.Thus, at these energies KTP can be considered as optically biaxial.Furthermore, as follows from considerations mentioned above, this coincidence of two singular optic axes takes place in a plane defined by two dielectric axes, i.e., in the a-c-plane for E ≈ 6.60 eV (point ii) and E ≈ 8.22 eV (point iv), as well as in the b-c-plane for E ≈ 5.85 eV (point i) and E ≈ 7.42 eV (point iii) (cf., Figure 2b).

Monoclinic Crystal Structure
In monoclinic crystals, there exists one nonorthogonal angle between two crystallographic axes and thus the dielectric tensor has one nonzero off-diagonal element.In the following, we assume that the symmetry axis is along the z-direction, i.e., in this system ε xy 6 ¼ 0 and ε xz ¼ ε yz ¼ 0 holds.Then, the orientation, the corresponding polarization, and the complex refractive index of two singular optic axes are connected with each other by Note, if the symmetry axis is within the x-y-plane, i.e., ε xz 6 ¼ 0 and ε xy ¼ ε yz ¼ 0 holds, the symmetry relations (Equation ( 18)) are slightly different.
As already mentioned, singular optic axes can coincide with each other and thus reduce the number of (singular) optic axes and/or form a classical (absorbing) optic axis. [8,11]In this case, the material can be optically triaxial (T ), biaxial (B), or uniaxial (U ).In the following, we denote (θ 0 , ϕ 0 ) and (θ 1 , ϕ 1 ), the polar and azimuth angle of the singular optic axis, with a corresponding eigenstate having a S 3 ¼ þ1 or S 3 ¼ À1 polarization, respectively.The orientation of the other two (singular) optic axes is determined by Equation (18a).The orientation of the (singular) optic axes for triaxial (T i ), biaxial (B i ), and uniaxial (U i ) is then: • T 1 : three singular optic axes In this case, the axes of one singular optic axes pair coincide with each other.This situation takes place if θ 0 ∈ 0, π f g and θ 1 ∈ 0, π f g, i.e., the singular optic axes coincide along the z-direction.
• T 2 : one classical and two singular optic axes For the formation of a classical optic axis, two singular optic axes with differently polarized eigenmodes have to coincide with each other.This situation can occur if all singular optic axes have the same azimuth angle, i.e., for θ 0 ¼ π=2 and θ 1 ∈ 0, π=2, π f g .• B 1 : two singular optic axes This situation takes place if the eigenmodes of all four directions in one hemisphere have the same polarization and the azimuth angles of the two axes pairs coincide with each other, i.e., ϕ 0 ¼ ϕ 1 and θ 1 ¼ π À θ 0 .
• B 3 : one singular and one classical optic axis In this case, the two directions of one pair of singular optic axes are antiparallel to each other, i.e., they are forming a classical optic axis, whereas the directions of the other two singular axes are parallel to each other.Thus, the polar angles have to be θ 0 ¼ π=2 and θ 1 ∈ 0, π f g. • U 1 : one circular optic axis All singular optic axes are parallel to the z-direction and the directions of differently polarized eigenmodes are pointing in the opposite direction, i.e., θ 0 ∈ 0, π f g and θ 1 ¼ π À θ 0 .• U 2 : one classical optic axis All singular optic axes are parallel to the z-direction and the differently polarized eigenmodes have the same direction, i.e., 1.) θ 0 ∈ 0, π f g and θ 1 ¼ θ 0 or 2.) θ 1 ¼ θ 0 ¼ π=2 and ϕ 1 ¼ ϕ 0 .Exemplarily for this symmetry class, the azimuth (ϕ) and polar angle (θ) of the singular optic axes for β-Ga 2 O 3 are shown in Figure 2c as a function of the energy.As mentioned above, due to the reduced symmetry of the monoclinic crystal structure compared to the orthorhombic one, only two (singular) optic axes are related to each other (Equation ( 18)), which is reflected by the twofold rotation symmetry of its spatial distribution around the symmetry axis.A mirror plane is present in the plane perpendicular to the rotation axis only if the polarization of the eigenmode of the corresponding direction is neglected.Thus, in total for the monoclinic crystal structure only one rotation axis and one mirror plane exist with respect to the spatial distribution of the (singular) optic axes.
In the presented spectral range, high symmetry points are present.In the energy range of E ≈ 7.08 eV : : : 7.15 eV, marked by i in Figure 2c, the symmetry point B 2 is almost achieved.In this spectral range, the polar angle for all four singular optic axes is almost π=2 and thus they are almost in the a-c-plane and, due to the symmetry relations (Equation 18), two axes with oppositely polarized eigenmode are close to each other.More precisely, at E ≈ 7.086 eV and E ≈ 7.135 eV, two eigenmodes coincide together forming a triaxial point T 2 , i.e., the presence of the one classical and two singular optic axes, but the other two singular optic axes are very close to each other.
Another triaxial point T 2 can be found at E ≈ 7.62 eV (marked by ii).An interesting situation occurs for energies larger than E ¼ 7.62 eV, indicated by the gray area in Figure 2c, where all directions with a circular right (left) polarized eigenmode are pointing in the z > 0 (z < 0) direction.For E ≈ 7.90 : : : 8.00 eV (marked by iii), two singular optic axes are almost in the a-c-plane and a triaxial point T 2 is nearly formed.

Triclinic Materials
In the case of triclinic materials, all three crystallographic axes are nonorthogonal to each other and thus all three off-diagonal elements of the dielectric tensor are nonzero, i.e., ε ij 6 ¼ 0. As a consequence of the low symmetry of the triclinic crystal structure, only Equation (15) holds.However, the orientations of the four axes are not fully independent of each other and the solutions of the characteristic polynomial (Equation ( 12)) are connected with each other, as recently shown by Grundmann et al. [28] This means the orientation of three axes determines the orientation of the fourth one.
Similar to the orthorhombic and monoclinic case, singular optic axes can coincide with each other, forming triaxial (T 1 and T 2 ), biaxial (B 1 , B 2 , and B 3 ), and uniaxial (U 1 and U 2 ) points.Note, the nomenclature of the special points is the same as for the monoclinic and orthorhombic case.A prediction of the angular position, as was done for the monoclinic case, is quite complicated as there is no simple relationship between the polar and azimuth angles of the singular optic axes.The conditions for the dielectric tensor for these special cases were discussed by Golovina et al. [12] The polar (θ) and azimuth (ϕ) angles of the singular optic axes in K 2 Cr 2 O 7 are shown in Figure 2e as a function of the photon energy, exemplarily for this material class.It can be easily seen that only Equation ( 15) holds.Furthermore, there is no axis of rotation or a mirror plane, as can be seen in the stereographic projection of the axis (Figure 2f ).Only an inversion center at 0, 0, 0 ð Þ T is present, if the polarization of the eigenmode is neglected.By comparing the spatial distribution of the orthorhombic, monoclinic, and triclinic crystals, it is evident that, besides the number of different refractive indices, [33] the crystal symmetry can also be distinguished by the symmetry of the spatial orientation of the four singular optic axes.Table 1 summarizes the relationship of polar and azimuth angles of the singular optic axes for different crystal structures, whereas an overview of the symmetry operations and the number of different refractive indices of the singular optic axes is given in Table 2.
In the investigated spectral range, an almost uniaxial point (U 2 ) is observable in the transparent spectral range at E ≈ 1.61 eV (marked by i), where two optic axes point in the same direction.A triclinic point T 2 can be found at E ≈ 3.46 eV (marked by ii).However, the other two singular optic axes are also very close to each other so that the material at this energy can be considered as almost optically biaxial (B 2 ).Interestingly, in the spectral range from E ≈ 3.27 : : : 3.47 eV (gray area), two axes with oppositely polarized eigenmodes are close to each other (the opening angle is less than 2°) so that the material in this spectral range behaves almost optically triaxial.In addition to this energy, almost triaxial points (T 2 ) can be also found at energies of E ≈ 4.92 eV and 6.10 eV, marked by iv and v in Figure 2e,f.An almost triaxial point T 1 can be observed at an energy of 3.60 eV (marked by iii), where two singular optic axes with same polarization as the eigenmode are close to each other (opening angle about 10 degrees).
The refractive index of the singular optic axes as a function of the energy is shown in Figure 3.As mentioned above, in the transparent spectral range the two optic axes have the same refractive index.As soon as the absorption sets in, each optic axis split into two singular optic axes and the corresponding refractive index differs for all four axes.However, in the case where the singular optic axes coincide with each other, the refractive index of the corresponding axes is degenerate, too.

Optically Active Materials
For some materials, an additional polarization dependence of the wave properties within the crystal takes place that cannot be explained by the dielectric tensor only.This effect is called optical activity and is typically known as circular birefringence and dichroism.If absorption is neglected, this leads to a change of the polarization plane through transmission through the crystal.Well-known materials, which are optically active, are quartz and sugar solutions. [34]n order to describe the properties of the wave in an optically active material, the constitutive equations have to be extended.It Table 1.Symmetry relations of the azimuth (ϕ) and polar (θ) angles for the directions of the singular optic axes having the same complex refractive index for the optically anisotropic crystals.Note, the coordinate system for the monoclinic crystals was chosen in such a way that ε xy 6 ¼ 0 and ε xz ¼ ε yz ¼ 0 holds.

Euler angle
Table 2.The number of (singular) optic axes (N a ) and different (complex) refractive indices (N n ) for these axes in the transparent (ε 2 ¼ 0) and absorption spectral range (ε 2 6 ¼ 0) for the different crystal structures.The last column indicates the symmetry operations of the angular distribution of the (singular) optic axes in the absorption spectral range. Crystal was already theoretically discussed [35][36][37][38] and experimentally shown [39] that only a symmetric set of constitutive equations can describe the properties of the wave within the crystal physically correctly.By using the approach proposed by Tellegen, the constitutive equation reads [38,40] with α being the gyration tensor.Whereas the crystal lattice determines the symmetry of the dielectric tensor, the corresponding point group determines the shape and magnitude of the gyration tensor α and thus, if the material is optically active or not.The wave equation in this case is then given by [41] 0 with s ij ¼ Àε ijk kk .The symbols ε ijk and kk represent the Levi-Civita symbol and the k th component of the wave vector k k ¼ 2π kk =λ, respectively.For α ¼ 0, the conventional wave equation for optically nonactive media is obtained.Interestingly, due to the presence of the gyration α, cones of singular optic axes can appear as shown by Merkulov. [42,43]An inspection of Equation (20) yields that if singular optic axes are present in the material, the presence of the optical activity does not lead to a change of the symmetry relations discussed above of these axes.This means, also for optically active materials, the spatial distribution of the singular optic axes is determined by the crystal lattice only.Furthermore, depending on the direction on which the optical activity is acting, the formation of (singular) optic axes can be suppressed.In the case of an optically isotropic material, e.g., cubic crystals or sugar solutions, the optical activity leads to circular birefringence and dichroism and thus the propagation properties depend on the polarization of the wave which is not the case for an optic axis.The same holds for crystals which belong to the 4 or 42m point group.These crystals have a tetragonal crystal structure and without optical activity an optical axis along the z-direction would exist.However, a wave propagating in this direction will "feel" the optical activity and thus the propagation properties depend on the polarization and an optical axis does not exist anymore.A summary of the shape of the dielectric tensor as well as of the gyration tensor, the number of (singular) optic axes, and different refractive indices for the different crystal symmetries and point groups are given in Table 3 and 4. Note, there are other symmetric sets of constitutive equations proposed in the literature, e.g., proposed by Voigt, [36] Drude, [37] and Post. [44]However, they will not change the symmetry relations as discussed above.

Similarities to EPs
For homogeneous plane waves, the real and imaginary parts of the wave vector point in the same direction and a propagation direction can be easily defined.As the (complex) refractive index of the forward and backward traveling wave is the same, the propagation direction forms an axis and it is convenient to name this axis a singular optic axis.However, as mentioned in the beginning, these singular optic axes can also be considered as EPs. [13]he term EP is typically used in non-Hermitian physics. [24,45]An inspection of Equation ( 7) or 9 yields that in the case of the propagation of homogeneous plane waves, the non-Hermetian Hamiltonian operator is represented by the "reduced dielectric tensor" (ε 0 defined by Equation ( 8)) or by the inverse of the dielectric tensor (ε 0À1 2Â2 ), and the eigenvector and eigenvalue correspond to the electric field or displacement and complex refractive index, respectively.Besides the simultaneous degeneration of the eigenvalue and eigenvector, a signature of an EP is that the eigenstates and eigenvectors exchange by encircling such EP in the parameter space.In the case of EPs in bulk materials, the parameter space is 2D because the propagation direction of a wave within a crystal can be described by two parameters, namely, the polar (θ) and azimuthal (ϕ) angle.
In Figure 4, the complex refractive index of the two eigenmodes, i.e., the eigenvalue, for a closed trajectory in the parameter space is exemplarily shown for β-Ga 2 O 3 for an energy of E ¼ 6.5 eV.The complex refractive index was ordered in such a way that neither its real nor its imaginary part exhibits a discontinuity along the trajectory.In the general case, after a roundtrip the initial value of the complex refractive index is recovered.However, if the encircled area contains a direction of a singular optic axis or rather an EP, the two complex refractive indices are exchanged.

Wave Equation
Up to now the propagation within the crystal was discussed.However, experimentally, the wave within the crystal is often Table 3.The structure of the dielectric function (ε) and the gyration tensor (α) for different crystal symmetries and point groups as well as the number of the (singular) optic axes (N a ) and the number of different refractive indices of these axes (N n ).The indices "oa" and "soa" indicate the presence of a classical or singular optic axis, respectively.
excited through an external medium, e.g., air.In this case, the wave has to pass an interface and the in-plane component of the wave vector, which is parallel to the interface, is conserved.
The corresponding out-of-plane component of the wave vector is then given by the wave equation.This conservation of the inplane component of the wave vector results in the planes of constant amplitudes and phases in the crystal generally no longer being coplanar, i.e., ℛek∦ℐmk, and the corresponding waves are called inhomogeneous or nonuniform plane waves. [27]In this case, the wave equation (Equation ( 1)) for the electric field and displacement can be written as [1,34] and respectively.If we write the wave vector by k ¼ 2π k=λ, where k can be termed as the refractive index vector since j kj ¼ ñ.The matrix s in Equation ( 21) is given by Table 4.The structure of the dielectric function (ε) and the gyration tensor (α) for different crystal symmetries and point groups as well as the number of the (singular) optic axes (N a ) and the number of different refractive indices of these axes (N n ).The indices "oa" and "soa" indicate the presence of a classical or singular optic axis, respectively. .
with ki being the component of k along the i-th direction.For a homogenous wave, k would correspond to the direction vector of the wave propagation and its magnitude to the (complex) refractive index (ñ).In the case of inhomogeneous plane waves, the real and imaginary part of k represents the direction for the planes of constant phase and amplitude, respectively.Equation (21)   requires that the dielectric function ε is given in the framework defined by the wave vector k.However, the dielectric function is typically given in the framework defined by the crystallographic axes (cf., Table 3 and 4), which is denoted in the following as ε c .
In this case, a coordinate transformation has to be performed similar to the case of a homogenous wave (Section 2), which is done by using Euler angles, i.e., where R ϕ, θ, ψ ð Þin the zyz notation is given by [46] R ϕ, θ, ψ Note that in contrast to the definition of a direction, the transformation of a coordinate system into another requires three instead of two Euler angles. [32]lthough Equation (21a) and (21b) represent a system of three coupled equations, only two of them are independent of each other because the electric field and the displacement are determined by two independent components only, as mentioned in Section 2.1.Thus, similar for a homogenous wave, the wave equation can be expressed by a system of two coupled equations only.In the following, we assume without loss of generality that the wave is propagating in the x-z-plane, i.e., ky ¼ 0. The corresponding wave equations for the electric field and the dielectric displacement can be then written as with Note, the circumflex ( ^) indicates that the vectors of the electric field and displacement consist of only the x and y components.The entire vector of the electric field and displacement is then given by E ¼ η E Ê and D ¼ η D D, respectively.
For a nontrivial solution of the wave equation, the determinant of the matrix M E and M D for the electric field and dielectric displacement, respectively, has to vanish.For a given in-plane wave vector component kx , this leads to a quartic function in kz .In general, this quartic function has four solutions, where two correspond to the forward and the other two to the backward traveling waves with respect to the z-direction.A degeneration of the two solutions can be found if the discriminant of this quartic function vanishes.
A special situation appears for an optically isotropic medium.In this case, the dielectric function reduces to a scalar function, i.e., ε ¼ ε ii and ε ij ¼ 0 for i 6 ¼ j.The wave equation for the electric field is then given by For the corresponding equation for the displacement, one has to replace the electric field Ê by the displacement vector D, only.As can be easily seen, for a given kx the solution of the wave function reduces to a quadratic function in k2 z and thus the two solutions of the forward and backward traveling waves are degenerate, and given by kz ¼ AE . Furthermore, the two independent components of the electric field, or rather of the displacement, are decoupled from each other and thus a basis of orthogonal states exists, independently if the transparent or absorption spectral range is considered.However, this is not the case for optically anisotropic materials, which will be considered in the following.

Biaxial Crystals
In order to find the crystallographic orientation where the eigenvalue and eigenvector are degenerate simultaneously for a given wave vector k, the dielectric tensor was transferred from the crystallographic system into the system defined by the wave vector, i.e., k ¼ kx , 0, kz T by means of Equation (23).In this case, for a given x-component of the wave vector ( kx ), the resulting wave equation is a quartic equation in kz in general with nonvanishing odd terms in kz .Note, in contrast to harmonic waves or waves in isotropic and uniaxial materials, kz of the forward and backward traveling wave differs not only in the sign of the real and imaginary part but also in their magnitude and therefore a degeneration of complex refractive index of the forward and backward wave does not necessarily take place.The corresponding crystal orientations for a degeneration of the wave vector and thus of the propagation properties of the waves, as well as of the eigenvectors, can be calculated by finding the zeros of the discriminant of the wave equation with respect to kz .Exemplarily for a selected energy and an in-plane wave vector component kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p , which would correspond to an injection of the wave in transmission or reflection geometry under an angle of incidence in air of 60°degree, the discriminant is shown for KTP (orthorhombic), β-Ga 2 O 3 (monoclinic), and K 2 Cr 2 O 7 (triclinic) in Figure 5 as a function of the Euler angles ϕ and θ for ψ ¼ 0. In contrast to homogenous plane waves, instead of eight orientations, which correspond to the orientation of the singular optic axes, 16 crystal orientations for ψ ¼ 0 are observable where the discriminant reaches a local minimum.This can be explained by the fact that, as mentioned above, in the case of inhomogeneous waves, a degeneration of the forward traveling waves ( kz > 0) does not necessarily coincide with a degeneration of the backward traveling waves ( kz < 0).
It should be mentioned that for another given Euler angle ψ, slightly different ϕ, θ ð Þpairs can be found so that there exists, in general, an infinite set of crystal orientations for a simultaneous degeneration of the propagation properties and polarization state.Exemplarily, the Euler angles ϕ and θ as a function of ψ are shown for triclinic K 2 Cr 2 O 7 at E ¼ 5.2 eV in Figure 6, for monoclinic Ga 2 O 3 at E ¼ 5.2 eV in Figure 9, and for orthorhombic KTP at E ¼ 3.6 eV in Figure 10 for a wave vector component kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p .For all materials, instead of eight well-defined crystal orientations, an infinite set or orientations exists, where two eigenmodes and eigenvalues are degenerated.However, these orientations can be grouped into 16 sets, 8 for the forward and 8 for the backward propagating waves.Similar to the case of homogenous waves, the symmetry of the crystal determines also the symmetry/relationship of the crystal orientation for a degeneration of the wave properties, which will be discussed in the following.Triclinic Crystals: For triclinic crystals in each case, four out of the 16 sets are connected with each other and the Euler angles of one subset determines those of the other three subsets.An inspection of the wave equation (Equation ( 25)) yields that if the Euler angle set ϕ 0 , θ 0 , ψ 0 ð Þleads to a degeneration of the eigenvalues and eigenmodes, then three other sets of Euler angles can be found, which are given by ϕ 0 , θ 0 , The wave vector component k z and the degree of circular polarization S 3 of these sets of Euler angles are then given by This symmetry relation can also be seen in Figure 7a.
The above used coordinate system is defined in such a way that the wave is propagating in the x-z-plane, i.e., ky ¼ 0, and the wave has a given kx .The advantage of considering this coordinate system is that the magnitude of kx can be experimentally set by using an appropriate angle of incidence (α) in reflection or transmission experiments.In this case, the wave component kx is given by kx ¼ n i sin α, with n i being the refractive index of the incident medium.Another natural coordinate system is the one defined by the crystallographic axes.The transformation between these two systems can be done by using the rotation matrix defined in Equation 24.As the directions of the real and imaginary part of the wave vector do not coincide with each other, i.e., ℛek∦ℐmk, we consider also in the following the direction of the wave time-averaged energy flux (p) given by and jpj ¼ 1 The directions of the real and imaginary part of the wave vector, as well as of the energy flux corresponding for a degeneration of the two waves, are shown in Figure6b,c in a stereographic projection for K 2 Cr 2 O 7 .For each quantity, four directions are observable.However, a closer look yields that these directions are not well-defined directions but rather represent a set of directions describing a cone around its barycenter (see inset of Figure 6b,c).Note, for the dielectric function of K 2 Cr 2 O 7 and kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p , the opening angle of the corresponding cone is quite small.Interestingly, whereas the cones for the real and imaginary part of the wave vector have a radial symmetry, this does not necessarily hold for the energy flux.Furthermore, the wave vectors for the forward and backward propagating waves are described by the same cone, so that in one hemisphere there can be four cones maximum for the degeneration of the eigenvalues and eigenvectors.Thus, four axes exist which are independent of each other, which is comparable to the case of homogenous waves where, for triclinic materials, four welldefined directions exist.However, whereas for homogenous Voigt waves these axes correspond to singular optic axes, in the case of inhomogeneous Voigt waves these axes correspond to the barycenter of cones of directions.Note, in contrast to singular optic axes for homogenous Voigt waves, the axes of the barycenters of the cones for the real and imaginary part of the wave vector and the properties of the waves, which belong to one cone, differ from each other and are a consequence of the inhomogeneous character of the wave.The change of the wave properties is exemplarily shown in Figure 7 for the wave vector component kz .Due to the symmetry relation given in Equation ( 29), the change of the properties for a circulation in clockwise direction of the cone for kz > 0 is the same as for a counterclockwise rotation for kz < 0. Furthermore, if ψ c,0 , θ c,0 represents the Euler angles for the direction of the barycenter of a cone, then another direction can be found for The presence of a set of directions for the wave vector for the degeneration of the eigenvalues and eigenvectors instead of a well-defined direction or rather axis can be understood in such way that the inhomogeneous character of the wave causes an additional impact on the polarization.If the wave vector is split into a homogenous part, where the real and imaginary parts are pointing in the same direction (k h ), and an inhomogeneous part (k inh ), i.e., k ¼ k h þ k inh , the wave equation can be written as A comparison with the wave equation for optically active materials (Equation ( 20)) exhibits a similar structure where the factor s inh is comparable to the optical gyration tensor.Thus, as for optically active materials, the presence of these contributions can lead to cones of singular optic axes, as is observed here.Note, in contrast to homogenous waves in optically active materials, only the matrix s x and not its transposed value is taken into account.
In contrast to homogenous Voigt waves, in the case of inhomogeneous waves, the degenerate eigenstates are not circularly polarized, which can be also explained by means of Equation ( 32) by considering the contribution of the inhomogeneous part, i.e., we write the wave equation as 0 As discussed in Section 2.1 and in refs.[5-8,28], for a complexvalued dielectric function a degeneration of the waves having a circular polarization can be found for ℋ 0 E ¼ 0. However, in order to fulfill Equation ( 34), V i E has to be zero, which is not the case in general.Thus, V i and therewith the inhomogeneous part of the wave vector can be interpreted as a "perturbation", which leads to an elliptical polarization.
It can be seen from Equation 34b that the magnitude of this "perturbation", and therefore the degree of elliptical polarization, depends on the magnitude of the inhomogeneous contribution of the wave vector.As mentioned above, the solutions of the wave vector for a degeneration of the wave in the case of a inhomogeneous wave form a cone in the momentum space.Since the inhomogeneous parts of the wave vectors along such a cone differ from each other, the degree of the elliptical polarization differs too.In Figure 8, the change of the elliptical polarization, as well as the fraction of the inhomogeneous contribution of the wave vector along the cone (jk inh j=jk h j) and the magnitude of the perturbation (kV i Ek=kV h Ek) are shown, with k ⋅ k being the Frobenius norm.
Monoclinic: Similar to the case of homogeneous Voigt waves, the enhanced crystal symmetry is also reflected by an enhanced symmetry of the crystal orientations and directions of the wave vector.For the Euler angles, still 16 sets are obtained; however, for monoclinic crystals, 8 of them are related with each other.This means, besides the symmetry relations given in Equation 28, it also holds that This symmetry relation can be seen in Figure 9. Note, Equation ( 35) is valid in the coordinate system where the symmetry axis is parallel to the z-axis, i.e., it holds that ε xy 6 ¼ 0 and ε xz ¼ ε yz ¼ 0. The additional relationship of the Euler angles leads also to an additional restriction for the direction of the wave vectors and instead of four independent cones of directions for the wave vector, only two, which are independent of each other, are obtained.For the barycenter of the cone, besides Equation (31) it also holds that Orthorhombic: For orthorhombic materials, it holds that ε ij ¼ 0 for i 6 ¼ j.In this case, only one independent set of Euler angles exists from which the other orientations can be deduced by means of Equation ( 28), ( 29), (35), and (cf., Figure 10).As there is only one set of Euler angles for the orientations, there exists also one set of wave vectors from which the other one can be deduced.The direction of barycenter of the cones are related to each other by Equation ( 31), (36), and The symmetry relations for the sets of Euler angles for the biaxial crystals are summarized in Table 5.

Uniaxial Crystals
For optically uniaxial crystals, the dielectric function can be represented by a diagonal matrix with two independent components.In the following, we choose the z-axis as the symmetry axis and thus ε 4).As the x-and y-axis are indistinguishable, the first rotation around the z-axis does not change the dielectric tensor and the coordinate transformation is independent of the Euler angle ϕ.Thus, for uniaxial crystals only, two Euler angles, namely, θ and ψ, are required in order to describe the rotation of the crystal in a given framework.In the configuration chosen here, the rotation matrix is then given by R The solution of Equation (21) or rather Equation (25) for the (complex) refractive index can then be written as . The change of the degree of circular polarization (black squares), the ratio between the magnitude of the inhomogeneous (k kinh k) and the homogenous (k kh k) part of the wave vector (k ¼ k h þ k inh , red line), and the "perturbation" (kV i Ek=kV h Ek, green solid line) along the cone.Note, the inhomogeneous part of the wave vector was chosen in such way that its magnitude is minimized, i.e., kk inh k ¼ minkk À k h k. [001] [001] [100] Figure 9.The same as in Figure 6 but for monoclinic Ga 2 O 3 (β-phase) at an energy of E ¼ 5.2 eV and kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p .a) The Euler angles φ and θ for the crystal orientation having a degenerate inhomogeneous plane wave with kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p ¼ sinπ=3 as a function of the Euler angle ψ. b,c) Stereographic projection of the direction of the real (filled squares) and imaginary (empty squares) part of the wave vector (b) and of the energy flux (c) in the crystallographic system.The Euler angles, which are connected to each other by the symmetry relations (Equation ( 28), ( 29) and ( 35)), are arranged as a 2 Â 4 block.The circle (dashed line) in (b) represents the orientation of the real (imaginary) part of the wave vector.
The solution of Equation (39a), which does not depend on the propagation direction, is typically related to the ordinary beam, whereas Equation (39b) is related to the extraordinary beam whose properties depend on the propagation direction in a uniaxial material.Thus, a degeneration of the (complex) refractive holds for the ordinary and extraordinary beam.
Inserting Equation (39a) into Equation (39b), the Euler angle ψ for a degeneration of the complex refractive index for a given inplane wavevector can be found by with and ξ ¼ signk z .A solution of Equation ( 42) is given by For an Euler angle ψ 0 which fulfills Equation (42), the angle θ is then given by Interestingly, the Euler angles ψ and θ and thus their dispersion depend only on ε ⊥ and not on the magnitude of the optical anisotropy or birefringence.However, Equation ( 41) is only valid if ε ⊥ 6 ¼ ε k and thus optical anisotropy is required for the presence of this degeneration of the eigenstates and eigenvalues.In the case of vanishing optical anisotropy, the eigenvectors are decoupled from each other and thus nondegenerate as discussed in Section 3.1.
The higher crystal symmetry compared to triclinic, monoclinic, and orthorhombic crystals leads to further symmetry restrictions.Instead of four cones of wave vectors in the crystallographic coordinate system, with degenerated wave properties, eight well-defined directions for the real and imaginary part are present in uniaxial systems, four in each for the forward and backward propagating waves.As the wave properties of the ordinary ray depends only on the magnitude of the in-plane wave vector component (Equation (39a)), in the case of a degeneration of the wave properties, the magnitude of the eight wave vectors is the same for all orientations and the components of the wave vector differ only in their sign.An inspection of Equation ( 39), (41), and (43) yields that the solutions for the out-of-plane Table 5. Symmetry relations of the Euler angles for the optically anisotropic crystals.The corresponding sign of the z-component of the wave vector and circular component of the Stokes vector with respect to the crystal orientation at ϕ, θ, ψ ð Þ , i.e., k 0 z ¼ k z ϕ 0 , θ 0 , ψ 0 ð Þand S 0 3 ¼ S 3 ϕ 0 , θ 0 , ψ 0 ð Þ , are given in the last two columns.Note, the coordinate system for the monoclinic crystals was chosen in such way that ε xy 6 ¼ 0 and ε xz ¼ ε yz ¼ 0 holds.N a denotes the number of sets of Euler angles which are related to each other.

Euler angle
component of the wave vector, as well as for the Stokes vector component S 3 , are connected to each other.In Table 5, the relationship is given.
In the following, we discuss the crystal orientations and wave properties for a degeneration of the eigenmodes for a uniaxial crystal, exemplarily for ZnO in detail.The discriminant of M E and M D (Equation 21) as a function of the Euler angles ψ and θ for an in-plane wave vector kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p is shown in Figure 11 in the transparent as well as in the absorption spectral range.In the transparent spectral range, two orientations can be found where the discriminant reaches zero, which corresponds to a degeneration of the wave vector.As the dielectric function is real-valued in the transparent spectral range, the argument of arccos in Equation ( 42) is real-valued, too, as well as larger than 1 and thus ψ ¼ nπ.Therefore, the degeneration takes place always for crystal orientations where the c-axis is within the plane defined by kx and kz .Furthermore, M E and M D of the wave equation (Equation ( 21)) are symmetric and real-valued and thus the corresponding eigenvalues are not degenerate.In contrast to this, in the absorption range there exist eight crystal orientations for a given in-plane wave vector, leading to a degeneration of the wave vector.Furthermore, as the dielectric function is complex-valued in this spectral range, the matrices M E and M D are non-Hermitian and thus the corresponding eigenvectors, i.e., the electric field and the dielectric displacement, are degenerate, too.This can be nicely seen in Figure 11c,d, where the circular polarization state (S 3 ) of the two eigenmodes is shown for the forward and backward propagating modes.It can be seen that there exist for the forward as well as backward propagating waves four crystal orientations where the eigenmodes are degenerated.Note, the Stokes vector was related within the framework system defined by the crystal, i.e., E c ¼ RE.In the case that the Stokes vector is related in the framework defined by the wave vectors kx and kz , the magnitude of the circular polarization slightly deviates from 1.
The dispersion of the Euler angles for selected in-plane wave vectors for ZnO is shown in Figure 12.The dispersion of the Euler angle ψ is similar to the imaginary part of ε ⊥ , whereas θ has a similar shape to the real part.This can be easily understood by the definition of ψ (Equation ( 43)), where the argument of the cosine function is given by ε ⊥ scaled by the in-plane wave vector.
Thus, abrupt changes in the dielectric function lead also to strong changes of the corresponding Euler angles.This can be nicely seen at the onset of the absorption.The strong change of the dielectric function caused by the excitonic contributions in this spectral range is reflected by a strong increase in ψ, and the angle θ exhibits a peak, similar to the real part of ε ⊥ .Interestingly, ψ is almost independent of the in-plane wave vector, if small wave vectors are considered.In contrast to this, an increase of kx increases the Euler angle θ, i.e., leading to an offset and to a more pronounced dispersion.The last one can be nicely seen in the vicinity of the bandgap, where the peak in θ is getting more pronounced with increasing kx .However, even for an in-plane wave vector of kx ≈ 1, the maximum change in θ is about 0.1π only.
The angles ψ and θ as a function of the dielectric function are shown in Figure 13 for kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p .Both ψ and θ are monotonic functions of the dielectric function and do not exhibit singularities.The maximum change of ψ and θ is about π=4 and π=3, respectively, in the presented range of the dielectric function, i.e., 1 < ε ⊥,1 < 8 and 0 < ε ⊥,2 < 8.As can be seen, the angle ψ is almost constant when the real and imaginary parts of the dielectric function change in the same way, i.e., ε 0 whereas in the case where they change in the opposite direction, i.e., ε 0 1 !ε 1 þ δ and ε 0 2 !ε 2 À δ, the angle ψ exhibits the largest change.This is in contrast to θ, which shows an almost radial dependence on the real and imaginary parts of the dielectric function.
From the experimental point of view, the Euler angle θ is often fixed and determined by the crystal orientation of the available substrate, e.g., the c-, r-, and m-plane orientation.In that case, the required in-plane wave vector kx and the azimuth angle ψ are of interest.The condition for a degeneration of the two waves for a given θ can be found for As ψ and therefore cos ψ have to be real-valued, the imaginary part of the right-hand side of Equation ( 45) has to vanish.Assuming real-valued kx , this is fulfilled for For a given energy, both ψ and kx show a monotonic increase with increasing θ and kx !∞ for θ !π=2.Exemplarily, the inplane wave vector kx as well as the Euler angle ψ are shown in Figure 14 as a function of the energy for ZnO.Again, the dispersion of the Euler angle ψ is similar to those of the imaginary part, whereas now the dispersion of the in-plane wave vector kx corresponds to the real part of ε ⊥ .Similar to the discussion for fixed kx above, with increasing θ the change of ψ and kx with respect to the energy increases and the increase of θ causes an offset in kx .Of particular interest are wave vectors with kx < 1 because waves with these in-plane vectors can be induced in a crystal by means of transmission and reflection measurements, due to the conservation of the in-plane moment at the interface.The relation between the angle of incidence α and the in-plane wave vector is given by kx ¼ n i sin α, with n i being the refractive index of the incident medium.In the case of ZnO, a degeneration of the eigenmodes with an in-plane wave vector kx < 1 can be achieved for crystal cuts with an Euler angle θ < 0.54π (cf., Figure 14).

EP in the Transparent Spectral Region
Up to now, the presence of EPs was discussed for small in-plane wave vectors, i.e., kx ≤ 1, so that these inhomogeneous waves can be induced from vacuum by means of reflectivity or transmission experiments where the in-plane component is given by kx ¼ n i sin α, with n i and α being the refractive index of the incident medium and the angle of incidence, respectively.However, a degeneration of the waves and thus EPs can be also present in the transparent spectral range for inhomogeneous waves if the wave vector, or one of its components, is complexvalued.This situation can appear if kx is complex-valued and therefore kz too, or, if kx is sufficiently large, so that kz becomes purely imaginary.In optically uniaxial materials, this situation Exemplarily, the crystal orientations for a degeneration of the wave properties are shown in Figure 15 for ZnO for a large in-plane wave vector component, i.e., kx ¼ 3.In contrast to the previously discussed cases with kx ≤ 1, instead of two, here four crystal orientations exist in the transparent spectral range, where the two eigenmodes are degenerate.Furthermore, the crystal is orientated in such a way that the crystallographic c-axis is within the kxky -plane.This can be easily seen in Equation (43) where the argument of arccos is now real-valued and the magnitude is smaller than one.Thus, this equation can be written as , which is purely realvalued, and ζ ψ ð Þ (Equation ( 41)), which is purely complex-valued.Similar to the case of homogenous waves, the crystal orientations for the forward and backward propagating waves are the same, which also follows from the symmetry relations given by Table 5.In the absorption spectral range (ε ⊥,2 6 ¼ 0), the argument of ffiffiffiffiffi ε ⊥ p = kx as well as ζ ψ ð Þ is complex-valued and thus θ 6 ¼ nπ and the crystallographic c-axis is not in the kxky -plane anymore and eight crystal orientations exist for the degeneration as is the case for small kx .The circular polarization component of the Stokes vector (S 3 ) is shown in Figure 15c,d for the transparent spectral range.One can see that in contrast to the case for small kx , for large kx the eigenmodes exhibit a strong degree of circular polarization even for crystal orientations where the waves are not degenerate.Only for orientations where two eigenmodes are degenerate, the waves are fully circularly polarized.
The crystal orientations as a function of the wave vector component kx for selected magnitudes of the imaginary part of the dielectric function are shown in Figure 16.As discussed above, for a transparent material, i.e., ε ⊥,2 ¼ 0, the angle ϕ ¼ nπ and θ increase monotonically from θ ¼ 0 for kx ¼ 0 with increasing kx until it reaches θ ¼ π=2 at kx ¼ ffiffiffiffiffi ε ⊥ p .Here, the crystallographic caxis is tilted within the kxkz -plane toward the x-axis.For larger kx , the c-axis is rotated in the kxky -plane, i.e., θ remains π=2 and ψ 6 ¼ 0. Note, for kx ¼ 0, ψ ¼ θ ¼ 0, the orientations are degenerate, i.e., the two eigenmodes form an orthogonal base and the propagation direction corresponds to an optic axis.In the absorption spectral range, i.e., ε ⊥,2 6 ¼ 0, ψ is nonzero for all in-plane wave vectors kx and increases with increasing ε ⊥,2 and increasing kx .Furthermore, for kx < ffiffiffiffiffi ε ⊥ p , ψ remains almost constant.The abrupt change of the crystal orientation at kx ¼ ffiffiffiffiffi ε ⊥ p , which is reflected by the steep onset of ψ and the constant value θ, smeared out with increasing ε 2 .For large kx , both ψ and θ reach π=2, i.e., the c-axis is along the y-axis.In this case, the directions coincide and an optical axis with an orthogonal base is formed.

Spatial Distribution and Symmetry Properties
If the thickness of the crystal is comparable to the coherence length of the light, the electric field still has to obey the wave equation, but the corresponding eigenmodes do not correspond to the eigenmodes of the material.In this case, the partial reflection and transmission at the interface (cf., Figure 17a) lead to interference of the propagating waves within the crystal and new modes are formed, which are named Fabry-Pérot modes.Furthermore, the two interfaces, which are parallel to each other, lead to a confinement of the out-of-plane component of the wave vector (k ⊥ ), i.e., k ⊥ is quantized.Thus, in contrast to the eigenmodes in a bulk and semi-infinite crystal, the mode energies for a given in-plane wave vector are discrete.In this case, the modes are distinguished by the mode number m.The presence of these modes can be nicely seen in the reflectance spectrum, where the minima are related to the modes.Such a (unpolarized) reflectivity spectrum is exemplarily shown for a crystal with a thickness of 1.1 μm in Figure 17b.Note, the case that is treated here is not limited to freestanding finite crystals only, but is also valid for (crystalline) films.As crystalline thin films are widely investigated, we will use in the following the term film also for crystalline films and finite plane-parallel crystals, and name the underlying material of the crystal, similar to a film, as the substrate.
The optical response of such films is typically described by using transfer matrices, [34,47] which can be written as with L i (L f ) being the interface matrix between the crystal and the upper (bottom) surrounding medium.P denotes the propagation matrix within the film.The vector Ξ i represents the electric field in the surrounding medium and can be written in the base of s-and p-polarized light as The thickness of the film is limited in the following discussion along the z-direction and the superscript þ (À) denotes the propagation of the wave in the positive (negative) z-direction.The eigenmodes in the film are then the nontrivial solution of Equation ( 47) for nonzero incident fields, i.e., refs.[16,48,49] with E s and E p being the amplitudes of the electric field for the sand p-polarized electric fields, with respect to the plane of incidence or rather emission, and J trans the Jones matrix for transmission.The last one can be calculated by means of the transfer matrix technique, [34,41,47,50] taking into account that (47).The solution of this field equation (Equation ( 48)) yields complex mode energies [53] So, whereas for a bulk and semi-infinite bulk crystal the eigenvalue corresponds to the refractive index, in the case of films and finite crystals, the eigenvalue is given by the complex mode energy.
In general, for a given in-plane wave vector and mode number, Equation ( 48) yields two solutions for a given mode number, which differ in their complex mode energy and corresponding eigenvector, which represents the polarization of the mode.In the case of an optically isotropic film or crystal, the Jones matrix is diagonal and the eigenstates correspond to s-and p-polarized electric fields as can be easily seen from Equation (48).For optically anisotropic materials the off-diagonal elements of the Jones matrix are nonzero and the two eigenstates are a linear combination of the s-and p-polarized electric fields.Similar to the wave equation, the Jones matrix in Equation ( 48) is a non-Hermitian matrix and so a simultaneous degeneration of the eigenmodes and the corresponding eigenvalue is possible.This means, in such a system, EPs can occur.
Exemplarily, for a 1.1 μm thick ZnO crystal surrounded by air, the difference of the energy, broadening, and complex mode energy is shown in Figure 18 for a mode number m ¼ 7. Similar to the bulk and semi-infinite crystals, there exist sets of in-plane wave vectors, where the energy and the broadening of the modes are degenerate, the so-called iso-energy and isobroadening lines in the momentum space, respectively (Figure 18a,b).The start (end) of such a curve for the degeneration of the energy (broadening) is the end (start) of the corresponding curve for the degeneration of the broadening (energy).At these points, the complex energy of the two modes is degenerate.
Besides the complex energy, the polarization of the two eigenmodes, corresponding to the eigenstate of Equation ( 48), also differs in general and is exemplarily shown in Figure 19 as a function of the in-plane wave vector for the 1.1 μm ZnO crystal shown in Figure 18.In most cases, the two eigenmodes are strongly linearly polarized.However, for a certain in-plane wave vector, the polarization of the two modes coincides with each other and the modes are fully circularly polarized.A comparison with the corresponding mode energy yields that the degeneration of the eigenstate is accompanied by the degeneration of the complex energy (eigenvalue) and thus these points represent an EP in the momentum space.This is also confirmed by the encircling of such a point (Figure 20).[56] In contrast to that, an exchange of the modes is not observable if the encircled area does not contain an EP.Note, the assignment of the modes was done in such way that neither the energy nor the broadening exhibits a discontinuity during the circulation.
The eigenmodes within the crystal occur due to the superposition of waves propagating in the forward and backward z-direction (cf., Figure 17a).Thus, the positions of the EPs in the momentum space are the same for positive and negative k z and only four EPs exist in the momentum space for a given mode number m, which are twofold degenerate.The corresponding in-plane wave vector at the EP can be written , where ϕ 0 is the polar angle with respect to the x-direction.From symmetry relations, it follows that the magnitude of the wave vector is the same for all four EPs and the position of two EPs are related to each other by ϕ 2 ¼ ϕ 0 þ π.If we choose the x-direction so that it coincides with the projection of one of the optic axes, then the orientation of the wave vector of the other two EPs is determined by The symmetry relations for the circular polarization are then given by In Table 6, these symmetry relations are summarized.Note, the superposition of the eigenmodes by modes having a positive and negative wave vector component k z leads to an enhancement of the symmetry of the system and thus, the spatial distribution of the EP does not depend on the crystal symmetry.

Properties of the EP
As mentioned above, due to the finite thickness of the crystal or film, the mode energy is quantized along the z-direction and characterized by the mode number m.Thus, in contrast to bulk or semi-infinite crystals, the EPs are not continuously but discretely distributed in the energy space.Furthermore, whereas EPs in bulk and semi-infinite crystals are an intrinsic property, their properties in finite crystals and films can be tuned by the  thickness as well as by the choice of the surrounding material.Exemplarily, the energy and broadening of the eigenmode with m ¼ 3 at the EP as a function of the thickness for a ZnO crystal are shown in Figure 21.Both the energy and the broadening decrease with increasing thickness, which can be attributed to a decrease in the quantization of k z .This is accompanied by a strong decrease of the energetic splitting between two consecutive modes, which decreases from 1.0 eV for a thickness of 300 nm down to 0.3 eV for a thickness of 1.1 μm.In contrast to the mode splitting, the difference of the broadening between two consecutive modes is not strongly affected by the thickness of the film.Note, the position in the momentum space of the EP is also independent of the film thickness.
The impact of the refractive index of the surrounding material on the energy and broadening of the mode at the EP is shown in Figure 22 for a 1.1 μm thick crystal.Here, the refractive index of the bottom material is varied, which mimics the situation that the crystal is attached to a substrate or that the film is deposited on different substrates.In this case, the energy of the mode at the EP exhibits a discontinuity and is reduced by 150 meV when the refractive index of the substrate matches that of the crystal.Besides this discontinuity, the energy is almost constant.In contrast to this, the refractive index of the substrate has a strong impact on the broadening of the mode.The broadening exhibits a singularity when the refractive index of the crystal matches those of the substrate.With increasing difference of the refractive index of the substrate and the crystal, the broadening of the mode decreases.Table 6.Symmetry relations for the appearance of the EP in a crystal with finite thickness.The change of the mode properties at the EP in dependence of the mode number m is shown in Figure 23 for the 1.1 μm thick ZnO crystal and selected refractive indices of the bottom material.The energy of the mode at the EP increases almost linear with increasing mode number m.This increase is independent on the refractive index of the surrounding material.In contrast to that, the broadening is almost independent on the mode number m and only a slight increase is observable.Similar to the energy, the magnitude of the corresponding wave vector (k k ) increases with increasing mode number m and the trajectory in the momentum space describes an almost straight line.As mentioned above, the position in the momentum space changes slightly with the refractive index of the bottom materials.
In order to understand the abovementioned behavior, the mode equation has to be solved.If we assume that the wave is propagating within the film with a refractive index n eff and an angle θ eff with respect to the normal of the interface, the mode condition can be approximated by [48,52,53] ra rs e 2i Ẽm n eff d f cos θ eff =ℏc ¼ 1 (52)   with ra (r s ) being the reflection coefficient of the film-ambient (substrate) interface, Ẽm the complex mode energy, and d f the thickness of the film.Note, the reflection coefficients are in the base of the eigenmodes, i.e., we assume for simplicity that there is no mode conversion.If we express the complex reflection coefficient of an interface by r ¼ re iϕ and the complex mode energy by Equation 49, the mode energy E m and broadening γ m are given by The index i ¼ 1, 2 denotes the two eigenmodes for a given mode number m, which are degenerated at the EP, whereas kk represents the magnitude of the reduced in-plane wave vector . Note, if the crystal is surrounded by the same material, then Φ ¼ 0. However, if the ambient material and the substrate consist of two different materials, the Brewster angles for the interfaces will be different.Thus, Φ is nonzero and can exhibit jumps between 0 and π when the in-plane wave vector crosses the Brewster angle of one interface.As can be seen from Equation 53a, these jumps in Φ lead to jumps in the mode energy (see Figure 22 and g B).
The out-of-plane component of the wave vector (k ⊥ ) is then given by which nicely demonstrates the quantization of k ⊥ .The energy and broadening calculated by Equation ( 53) as a function of the thickness are shown in Figure 21 by red solid lines, which agrees well with those obtained by solving the Maxwell equations (Equation ( 52)).From the decrease of the energy, the value of n eff cos θ eff was determined to be 1.975, which is between the refractive index of the ordinary and extraordinary wave of 1.991 and 2.000, respectively, in ZnO.The same value for n eff cos θ eff is obtained for the description of the splitting between two consecutive modes, which is given by and for the change of the mode energy as a function of the mode number for a given thickness (Figure 23a).A closer look yields, for large mode numbers, a small deviation compared to the results of the rigorous calculation of the Maxwell equation.This can be attributed to the increase of the in-plane wave vector  53).Due to the small in-plane wave vector of the mode at the EP, the reflectivity in Equation (53b) was calculated for simplicity for normal incidence assuming an isotropic refractive index of the crystal of n eff = 2.The vertical gray dashed line represents the refractive index of the ZnO crystal.The discontinuity occurs at n eff = n s , i.e., the reflection coefficient changes its sign and thus Φ changes from 0 to π.The change in the energy corresponds to πℏc=2d f n eff cos θ eff ≈ 150 meV.
component with increasing mode number, which will be discussed below.If this is considered (red dashed line), the increase of the photon energy is well described by (Equation (53a)).The broadening of the mode at the EP is mainly determined by the thickness and the refractive index of the film and the reflectivity of the interfaces (Equation (53b)).From the analysis of the decrease of the broadening with increasing thickness, the reflectivity of the interfaces was estimated to be about R ¼ 10%, which is comparable to those expected for a ZnO-vacuum interface close to normal incidence.Note, the small in-plane component of the wave vector (cf., Figure 23) indicates a small angle of incidence of this wave at the interface.The value for n eff cos θ eff was taken from the analysis of the photon energy.
As mentioned above, k k at the EP reveals an almost linear increase with increasing mode number, in the magnitude of the wave vector and in the k-space (Figure 23c,d).In order to explain this behavior, we have to recall that the broadening of the eigenmodes is almost independent of the mode number and therefore the iso-broadening lines in the momentum space, too.In contrast to this, the energy of the eigenmodes depends linearly on the mode number, and the energetic splitting between consecutive modes for a given in-plane wave vector is determined by the refractive index of the corresponding eigenmode (cf., Equation ( 55)).As the refractive index n eff ,i differs slightly for the two eigenmodes for a given mode number m, the energetic splitting between consecutive modes is different for the two eigenmodes, leading to a change of the iso-energy line in the k-space for different mode numbers.Furthermore, as it holds n eff ,1 > n eff ,2 for E 1 < E 2 this iso-energy line shifts to larger in-plane wave vectors with increasing m and therefore the crossing of the iso-energy and iso-broadening lines, i.e., the wave vector of the EP increases, too.As a further consequence, the wave vector of the modes at the EP describe almost a line in the k-space.
The enhancement of k k at the EP is also responsible for the slight increase in the broadening (cf., Figure 23b).With increasing in-plane wave vector, the reflectivity for the s-(p-) polarized light increases (decreases) for k vectors below the Brewster angle, as is considered here.At the EP, the mode is circularly polarized (Figure 19), which can be expressed by linear combination of the s-and p-polarized light.Typically, the increase in the reflectivity of the s-polarized mode is smaller than the decrease for the c) The magnitude of the in-plane wave vector component as well as d) the position of the EP in the momentum space.The optical axis was set to be parallel to the interface and along the x-direction.For simplicity, the refractive index was set to be constant for the whole spectral range.The refractive index of the underlying material, which corresponds in the case of a film to the substrate, is denoted as n s whereas the refractive index top material was set to vacuum (n a ¼ 1.0).The red solid lines in (a) and (b) represent the calculated behavior by means of Equation ( 53), assuming n eff and k k to be independent of the mode number, whereas in the case of the red dashed line in (a), the change of k k was taken into account.
p-polarization and thus the averaged reflectivity decreases causing the slight increase of the broadening.The situation is slightly different if one considers the impact of the reflectivity of the interface, i.e., the surrounding material of the film, on the wave vector of the EP for a given mode number m.Although the refractive index of the surrounding material has a strong impact on the broadening of the modes, the difference of the broadening between the two modes for a given wave vector is only slightly affected as the change in the difference of the broadening between the two modes can be estimated by Here, ΔR i ¼ R i,n s,1 =R i,n s,2 is the change in the reflectivity of the interface for two different surrounding materials of the two modes.Typically, ΔR 1 ¼ ΔR 2 and thus Δ Δγ ð Þ ≈0.Therefore, the iso-broadening does not strongly depend on the reflectivity of the interfaces.Furthermore, the energy of the mode and therefore the iso-energy line also do not depend on the reflectivity so that the wave vector of the EP is only slightly changed by the reflectivity.

Remarks: Experimental Observation
The experimental observation of the simultaneous degeneration of the energy and polarization might be smeared out due to the finite spot size, imperfect matching of the conditions for the EP, and experimental imperfections.Thus, experimentally, the presence of the EP is often proven by an encircling of this point and observing the exchange of the eigenmode.However, in the case of films this exchange of the eigenmodes is quite difficult to observe.This can be attributed to the small energy difference of the eigenmodes compared to the broadening, leading to a strong energetic overlap between these two modes.
In optically isotropic samples, typically the free spectral range between two consecutive modes compared to the broadening of the mode, the so-called finesse, is sufficient in order to estimate the overlap of the modes.However, in optically anisotropic samples, the situation is slightly more complicated because for each mode number two eigenmodes exist, which are spectrally not degenerate.Thus, the intermodel and intramodal free spectral range, i.e., the energetic splitting between consecutive mode numbers E FSR Δm À Á and between two eigenmodes of the same mode number E FSR m ð Þ, respectively, have to be considered.In order to spectrally resolve the eigenmodes, the free spectral range has to be compared to the broadening of the modes as well as to the energetic splitting between the two eigenmodes.The latter will be named in the following as the normalized free spectral range (G).The intramodal (ℱ m ), intermodal (ℱ Δm ) finesse as well as the normalized free spectral range are then given by where is the average broadening for a given mode number m, respectively.By means of Equation ( 53), these quantities can be approximated by Here, the modes were labeled in such a way that Typically, Δn ( 1 and the intermodal finesse (ℱ Δm ) and the normalized free spectral range (G) are larger than 1.However, the intramodal finesse is proportional to Δn, the difference of the effective refractive index for the two modes, and thus is much smaller than 1.For the ZnO film discussed here, the finesse is shown in Figure 24 as a function of the mode number m for selected surrounding materials.For the freestanding crystal, the intramodal finesse is about 0.06 for m ¼ 3 and increases linearly to 0.15 for m ¼ 9, which indicates a strong spectral overlap between the two eigenmodes.The finesse decreases if the refractive index of at least one surrounded material increases, which can be attributed to an increase in the broadening of the mode caused by the decrease in the reflectivity of the corresponding interface.
In principle, the intramodal finesse can be enhanced by increasing the mode number.However, there are still some practical limitations.The mode energy increases almost linearly with increasing mode number (cf., Figure 23).This would shift the mode energy into the UV spectral range where most of the materials are no longer transparent anymore or the experimental setup is incapable of measuring.A promising material might be α- [57][58][59][60] or β-Ga 2 O 3 , [31,33,61] which has a bandgap of about 4.8 eV.However, even in that case the mode number can be roughly double compared to the case considered here, and the intramodal finesse is still much smaller than one.In the visible spectral range, the mode number can be enhanced by increasing the thickness of the film.However, this is accompanied by a decrease in the broadening and the splitting between two consecutive modes (cf., Equation ( 55)) and would require a spectral resolution of a few μeV.A further possibility to enhance the intramodal finesse would be the use of materials having a large birefringence because ℱ m ∝ Δn and the difference of the effective refractive index between the two modes (Δn) increases with increasing birefringence.However, whereas (ℱ m ) is enhanced, the normalized free spectral range (G) decreases.
An inspection of Equation ( 58) yields that the intra-and intermodal finesse can be enhanced, by keeping the normalized free spectral range G almost unchanged and larger than one and by increasing the reflectivity of the interfaces.For sufficiently large reflectivity values, the film must be embedded between two mirrors.Note, changing the reflectivity of one interface is not sufficient because in this case the broadening of the modes will be limited by the interface exhibiting the lowest reflectivity.The presence of EPs in films embedded between two mirrors, also known as microcavities or microresonators, was first realized by Richter et al. in 2019 in a ZnO-based microresonator [18] and 2 years later in a lead-halide perovskite-based microresonator. [19]The general behavior of the EP in such microresonators is similar to those discussed in this section.As microresonators are of high interest, a detailed discussion with respect to the EP is given in the appendix (Appendix C).

Propagation
For the description of the wave properties, e.g., the propagation of the wave within the crystal or its reflection at an interface, it is quite common to describe the electric field or the corresponding displacement by a linear combination of the eigenstates of the system, see, e.g., refs.[1,34].As mentioned above, in the case of the singular optic axis, only one eigenstate exists and a description of an arbitrary polarization state or electric field by a linear combination of two states is not possible.Thus, the propagation of an arbitrarily polarized wave along the singular optic axis cannot be described with this formalism.This holds also for inhomogeneous waves, where for a given complex-valued wave vector the eigenmodes can be degenerate in their polarization state as well as in their propagation properties.This led Voigt to the conclusion that only an eigenmode can propagate with this wave vector along a singular optic axis and the other polarization states are forbidden.However, as Pancharatnam pointed out in 1955 by means of geometrical considerations using the Poincare sphere, [9] a homogeneous wave which does not have the polarization of the eigenmode can propagate along such a singular optic axis but will undergo a change of the polarization until the polarization coincides with that of the eigenmode. [10]In order to describe the propagation of the electromagnetic wave for all polarization states where the eigenmodes are degenerate, two strategies were proposed for homogeneous plane waves: 1) using an amplitude which depends on the spatial position [62,63] or 2) using a propagation matrix. [62]In the following, the two strategies will be discussed in detail and extended to inhomogeneous waves.Thereby, we assume the wave is propagating in the x-z-plane, i.e., the fields do not depend on the spatial position in the y-direction.

Spatially Dependent Amplitude
Typically, the wave equation (Equation (7) or rather Equation ( 9)) is solved by means of a plane wave, i.e., A r, t ð Þ ¼ A 0 e i krÀωt ð Þ with A being the field vector and k, ω are the wave vector and the angular frequency, respectively.The vector A 0 represents the amplitude and does not depend on the spatial position.In doing so, the wave equation typically yields two solutions.However, for welldefined wave vectors, which represent the propagation direction in the case of homogeneous waves, the solutions of the wave equation are degenerate as discussed in Section 2. In this case, an arbitrary polarization cannot be described by a linear combination of the eigenmodes and thus only the eigenmode can propagate along such a direction, which is in contradiction to the experiment. [10]Mathematically it can be shown that a differential equation of the type of the wave equation has, in the case of a simultaneous degeneration of the eigenmode and eigenvalues, a second solution where the amplitude depends on the spatial position. [63,64]For inhomogeneous waves, a solution of the wave equation for the electric field is then given by where kx and kz are related to the x-and z-components of the wave vector k, i.e., k ¼ k 0 kx , 0, kz T .The amplitude ℰ i is then determined by  (58a).Due to the small wave vector, the reflectivity at normal incidence was considered and an isotropic refractive index of the cavity was considered for the reflectivity, i.e., n eff ¼ 2. The refractive index of the ambient material in (a) and (b) was set to that of vacuum (n ¼ 1).The vertical gray dashed line in (b) denotes the refractive index of the crystal.
με À Equation (60a) corresponds to the wave equation and thus ℰ 1 corresponds to the polarization of the degenerate eigenstate, i.e., the polarization of the inhomogeneous Voigt wave.The general solution describing the polarization of an arbitrarily polarized wave for a wave vector k with a degenerate eigenmode is then given by The coefficients c 1 and c 2 are determined by the initial polarization.

Homogeneous Waves
In the case of homogeneous Voigt waves, the coordinate system can be chosen in such a way that the wave is propagating along the z 0 -direction, i.e., kx ¼ 0. As mentioned in Section 2.1, the wave equation can be reduced to a 2D problem and Equation (60) simplifies then to.
with ε0 being the reduced 2 Â 2 dielectric tensor defined by Equation ( 8) for a wave propagating along the z 0 -direction.Note, the vectors ℰ i are 2D vectors, i.e., they depend only on the x 0 -and y 0 -components of the electrical field, which are independent of each other.The full vector for ℰ i and thus E can be obtained by means of Equation (6).Note also that for the dielectric displacement a similar equation is obtained by replacing ℰ i in Equation ( 61) and ( 62) by Di and ε0 by ε 0À1 2Â2 ½ À1 , which is defined by Equation (9).From the Maxwell equation ∇D' ¼ 0, it follows that D z 0 ¼ 0. A similar solution for the dielectric displacement for homogeneous waves propagating along singular optic axes was presented by Borzodv. [62]

Propagation Matrix
One of the commonly used formalisms describing the propagation of electromagnetic waves by means of matrices was introduced by Jones in his famous series starting in 1941. [65]lthough this concept is very powerful in general, it breaks down if the eigenmodes are degenerate along the propagation direction because the Jones formalism requires the presence of two eigenmodes.Another widely used formalism was proposed by Berreman, which does not require the knowledge of the eigenstates and their corresponding propagation properties. [47]owever, in this formalism the description of the forward and backward traveling waves are simultaneously taken into account and thus an investigation of the forward or backward traveling waves only is quite difficult.For the propagation of homogeneous waves along a singular optic axis, Borzdov introduced a 2 Â 2 propagation matrix, [62] which will be extended in the following to inhomogeneous waves.
In general, the wave equation represents a system of three coupled equations for the field components.However, taking into account the constitutive equations, only two of the three equations are linearly independent.This is reflected by the fact that two of the three components of the electric field or dielectric displacement vector are independent of each other and the third one, often chosen as the z-component, can be expressed by a linear combination of the other two.As mentioned above, we assume for simplification and without loss of generality in the following that the wave is propagating in x-z-plane, i.e., ∂ y ¼ 0, and that the change of the phase in time and along the x-direction can be described by a scalar function, i.e., , with kx being a scalar representing the magnitude of the x-component of the wave vector and ω the angular frequency.The last assumption can be motivated by the fact that inhomogeneous waves are often excited from air like it is the case for transmission and reflection experiments and the in-plane momentum has to be conserved.If we use D ¼ εε 0 E and express E z by means of E x and E y , we obtain where ∂ z represents the differential operator along the z-direction.The wave equation can then be reduced to a coupled system of two equations given by with Note, Ê is a 2D vector representing the x-and y-components of the electric field.
A solution of Equation ( 64) is given by the ansatz with Ê0 being the amplitude at z ¼ 0 and k 2Â2 z a 2 Â 2 matrix.This is in contrast to the conventionally used ansatz where k z is a scalar and represents the z-component of the wave vector.Inserting Equation (66) into Equation ( 64), one obtains Equation ( 67) has to be fulfilled for all electric fields and thus a solution for M is given by M ij ¼ 0 which leads to a system of four coupled equations in k2Â2  63, the z-component of the electric field, as well as the propagation matrix for the entire three-dimensional electric field vector, is given by The change in the electric field is then given by For the dielectric displacement, the change in space and time can be described similar to Equation (69).However, whereas kx is the same for the electric field and dielectric displacement, the matrix kz differs between the two fields.Rewriting the wave equation as a function of the dielectric displacement and expressing D z by D x and Dy, the corresponding system of equations, similar to those for the electric field (Equation ( 67)), are given by As for the electric field, the eigenvectors and eigenvalues of kD,2Â2 z correspond to the eigenmode of the dielectric displacement and the corresponding out-of-plane component of the wave vector ( kz ), respectively.Thus, for the physical solution of the forward (backward) travelling waves the real and imaginary parts of the eigenvalues have to be positive (negative).By knowing k D,2Â2 z , the entire 3D vector of the As the electric and the displacement fields are connected with each other by the dielectric function, i.e., D ¼ εε 0 E, the corresponding propagation matrices describing the change along the z-direction are not independent of each other and are connected via.
Note, the matrix k3Â3 z is connected to k2Â2 z by Equation (68b) so that kD,2Â2 z and kD,3Â3 z can be also expressed by k2Â2 z .A special case occurs when the off-diagonal elements of the dielectric tensor vanish.In this case, A 0 and A D 0 in Equation ( 67) and (70), respectively, have a diagonal form and ) also has a diagonal form and is given by where þ (À) corresponds to the forward (backward) travelling wave.Note, the diagonal form of k2Â2 z ( nD,2Â2 z ) reflects the well-known fact that in this case the two eigenmodes are polarized along the y-direction and in the x-z-plane.The corresponding diagonal elements correspond to the in-plane moment kz , representing the eigenvalues of this matrix.

Homogeneous Waves
For the description of homogeneous waves, Equation ( 67) and (70) can be simplified.If z 0 represents the propagation direction of the wave, then The ' denotes the dielectric tensor for the propagation along the z 0 -axis and the þ (À) corresponds to forward (backward) travelling waves.
The 3 Â 3 propagation matrix for the dielectric displacement is then given by kD,3Â3 ¼ 0 has to be fulfilled.
In the case of the electric field, the propagation matrix is given by The propagation matrix mentioned above is defined in the framework by the propagation direction, i.e., the wave is propagating along the z 0 direction.In order to describe the propagation in the x-z-system, the propagation matrices have to be transformed accordingly.The propagation matrices in the x-z-system are then given by kx ¼ ÀR À1 kz 0 R sin φ (80) with R being the rotation matrix.Note, we assumed that the wave propagates within the x-z-plane and thus the electric field and the dielectric displacement do not depend on y.

Change of the Polarization State
By means of the two formalisms presented above, the propagation of the electromagnetic wave with an arbitrary polarization along a singular optic axis can be studied.Exemplarily, the intensity and the change in the polarization for a wave propagating along a singular optic axis in β-Ga 2 O 3 at 4.6 eV are shown in Figure 25 for selected initial polarizations.For a polarization of the wave, which corresponds to the eigenmode of the singular optic axis, the polarization of the wave does not change for propagation along the singular optic axis (Figure 25) and the degree of circular polarization is conserved for all z, as expected.Furthermore, the amplitude of the wave decays exponentially, determined by the imaginary part of kz .
If the wave does not have a polarization that belongs to the eigenmode, a change in the polarization occurs (Figure 25).Besides using geometrical considerations presented by Pancharatnam, [9,10] this change can be easily understood in the framework discussed above.1) By using a spatially dependent amplitude, the coefficient c 2 of the general solution (Equation ( 61)) is in this case nonzero, which causes a change in polarization along the propagation direction.Or 2) in the case of using a propagation matrix, the polarization does not coincide with the eigenvector of the matrix and thus depends on z.Interestingly, in contrast to the eigenmode, the amplitude of the wave does not decay exponentially due to the additional polarization rotation term given by c 2 6 ¼ 0.

Reflection and Transmission
For the calculation of the reflection and transmission coefficients at an interface, the incident, reflected, and transmitted waves are typically described by a linear combination of the two eigenmodes in the corresponding media.However, if only a single eigenmode exists for the given in-plane wave vector of the reflected and/or transmitted wave, this linear combination approach fails.In the classical approach, the polarization of this wave is then determined by the polarization of the eigenmode.However, as soon as the polarization of the reflected and/or transmitted wave is fixed, the calculation of the reflection and transmission coefficient will fail for the general case.Thus, for a singular optic axis perpendicular to the crystal surface, Voigt concluded that an incident wave at normal incidence is totally reflected, if its polarization does not match with that of the eigenmode of the singular optic axis. [5]But as was discussed in Section 5, waves with a polarization different to that of the eigenmode can propagate along a singular optic axis and therefore can enter into the crystal.By means of the formalism presented in Section 5, the corresponding transmission and reflection coefficients can be easily calculated.A possible solution thus might be to describe the electric field in the case of a degeneration of the eigenmodes by means of a spatially dependent amplitude (Section 5.1) or by means of a propagation matrix (Section 5.2).The last approach will be used in the following.This formalism does not require a special treatment of the degeneration of the eigenmodes and is more convenient for the general case.Again, we assume without loss of generality that the wave propagates in the x-z-plane and the z-direction is perpendicular to the interface.
For the calculation of the transmission and reflection coefficients, besides the electric field also the magnetic field has to be known, which is related to the electric field by ∇ Â E ¼ ÀB : ¼ Àμμ 0 H : .If we describe the time dependence of the magnetic field by H r, t ð Þ ¼ H r ð Þe iωt and express the z-component of the electric field by E x and E y by means of Equation (68a), the in-plane components of the magnetic field are given by At the interface, the in-plane components of the electric and magnetic fields are conserved, i.e., are fulfilled.The indices i, r, and t denote the field and the corresponding matrices of the incident, reflected, and transmitted wave, respectively.The reflected and transmitted field can be then expressed by the reflection and transmission matrices given by Similar to the Jones matrix typically used in ellipsometry, the entry of r (t) represents the reflection (transmission) coefficient r ij (t ij ) for a given component i into the component j.Equation (85a) and (85b) are valid in general and can be applied in the case of degenerate and nondegenerate eigenvalues and eigenstates.If the dielectric tensor of the incident medium has zero off-diagonal terms, then m r ¼ Àm i .
For the special case that the dielectric tensor in the considered coordinate system consists only of diagonal elements, the reflection and transmission matrices also have off-diagonal elements equal to zero.Thus, no mode conversion takes place, i.e., no electric field will be reflected (transmitted) into another component, as expected.The coefficients are then given by r ¼ r pp 0 0 r ss (86a) where kz is defined by Equation (76) and the superscript i and t represent the incident and the transmitted medium, respectively.]

Remarks
The presence of an interface in semi-infinite and finite crystals has a significant impact on the EP, their spatial and spectral distribution, as well as on the corresponding waves at the EP, which will be highlighted here.
As mentioned above, EPs in crystals and films arise due to the simultaneous degeneration of the eigenvalues and eigenstates, which requires, from the mathematical point of view, a non-Hermitian matrix.For bulk and semi-infinite crystals, this matrix is given by the wave equation and the eigenvalues correspond to the (complex) refractive index, whereas in the case of finite crystals and films, the eigenmodes are a superposition of forward and backward propagating waves.In this case, the eigensystem is represented by the transfer matrix and the eigenvalues are represented by the complex mode energy.
In order to have a non-Hermitian character of the corresponding eigensystem, the presence of absorption, i.e., dissipation, is required in bulk crystals.In this case, the complex-valued dielectric function causes a complex-valued wave equation.For semiinfinite and finite crystals, the situation is a little bit different.In the absorption spectral region, the corresponding eigensystem for these two systems is complex-valued, too, and EPs can appear.However, in addition to this, EPs can also appear in the transparent spectral region.For semi-infinite crystals this situation can appear if the in-plane wave vector is sufficiently large, i.e., if one component of the wave vector is complex-valued.For uniaxial materials, this is the case if k2 k > ε ⊥ .In this case, the amplitude front decreases perpendicular to the interface.For finite crystals, dissipation is provided by the interfaces, which leads to a partial transmission of the electromagnetic wave into the environment and thus to a coupling to the vacuum states.Furthermore, EPs in bulk crystals appear only for optically biaxial materials, i.e., crystals having a triclinic, monoclinic, and orthorhombic crystal structure, whereas for semi-infinite and finite crystals these EPs can also occur in optically uniaxial materials, i.e., in materials with a hexagonal, trigonal, or tetragonal crystal structure.
The presence of the interface also has an impact on the distribution of the EPs.In contrast to the propagation in bulk crystals, the symmetry of the distribution of the EPs in the momentum space in finite crystals is independent of the crystal symmetry.Thus, Equation ( 50) and ( 51) are valid independently of the crystal symmetry of the finite crystal or film.Exemplarily, the difference of the complex mode energy in the momentum space is shown in Figure 26 for orthorhombic KTP, which yields the same symmetry as for the ZnO crystals discussed in Section 4. The independence of the crystal symmetry can be easily understood by the fact that the dielectric function of triclinic and monoclinic crystals can be diagonalized for a given energy in the transparent spectral region and thus have the same shape as for an orthorhombic material.On the other hand, as stated above, the eigenmode is a "superposition" of a forward and backward traveling wave.If we neglect the polarization of the eigenmode, a finite crystal or thin film made of a uniaxial material exhibits three C 2 symmetry axes and two mirror planes.Thus, the spatial distribution of the EPs exhibits the same symmetry as an orthorhombic crystal.
The properties of the EP in bulk crystals are determined by the dielectric function and thus are an intrinsic property of the corresponding material.In contrast to this, for crystals with at least one interface, the properties of the wave at the EP can be controlled.For semi-infinite crystals, the properties are determined by the magnitude of the in-plane wave vector component, which can be adjusted by a coupling of an electromagnetic wave from the vacuum into the crystal.For finite crystals and films, the properties of the EPs for a given crystal can be further adjusted by the thickness of the crystal and the refractive index of the surrounding materials.Furthermore, here the EPs are discreetly distributed in the energy space and not continuously as for the bulk and semi-infinite crystals.
Due to wave-particle duality, the electromagnetic waves in the crystal can be described as particles, photons.For bulk and semiinfinite bulk crystals, photons are massless particles.However, due to the vertical confinement of the wave vector (k z ) in finite crystals and films, the corresponding photons are not massless particles anymore but have a finite mass.For small in-plane wave vectors, Equation 53a yields a parabolic dispersion and in the vicinity of k k ≈ 0 the mass is given by eff being the ground state energy of the mode with mode number m.For the modes in the nearinfrared up to the near-UV spectral range, the photon mass is on the order of 10 À5 m e , with m e being the rest mass of an electron.
The eigenmodes of the waves at the EPs are fully circularly polarized and an emission of either left-or right-handed circularly polarized light from such an EP would be expected.However, for microresonators an emission of elliptically polarized light was experimentally observed. [18,19]Besides experimental limitations, e.g., finite spot size, this can be attributed on the one hand to the fact that the energy of the eigenmode is complexvalued, whereas in the experiment a real-valued energy is measured.On the other hand, the polarization of the eigenstate is determined within the finite crystal or film.For the emission into the surrounding medium of the optical system, the electromagnetic wave has to be (partially) transmitted through the interfaces of the film or through the mirror in the case of the microresonator.This is determined by the transmission coefficient.Similar to the reflection coefficient, the transmission coefficient depends on the polarization, which is defined by the interface.Polarization parallel and perpendicular to the interface suffer different transmission and a circular polarization is, in general, not conserved, leading to a conversion of the circular polarization of the emitted wave into an elliptical one.A summary of the discussed differences in crystals with and without interfaces is given in Table 7

Summary
In optically anisotropic crystals, two eigenmodes exist, which differ in general by their eigenstate, namely, their polarization and their propagation properties.However, for some distinct propagation directions, both the eigenstate and the propagation properties can be degenerate at the same time.For optically biaxial bulk crystals, this situation occurs along the so-called singular optic axis.More generally, this situation corresponds to an EP in the momentum space.Here, we have shown that the presence of EPs in the momentum space is an intrinsic property of the optically anisotropic crystal and occurs due to the non-Hermitian nature of the wave equation or transfer matrix, reflecting the competing interaction between the anisotropic propagation properties and losses.In the case of bulk and semi-infinite crystals, the losses are caused by the absorption within the crystal, whereas in finite crystals the losses are caused by the partial reflection and transmission of the wave at the interfaces.The crystal symmetry has a strong impact on the spatial distribution of the EPs in momentum space.Whereas for orthorhombic crystals the spatial distribution of all EPs is connected with each other, for triclinic materials the distribution of the EPs is almost independent of each other.As a consequence, the crystal symmetry also has an impact on the degeneration of EPs in these materials, i.e., the presence of uniaxial, biaxial, and triaxial points and the simultaneous presence of singular optic and classical optic axes.
The presence of an interface has a strong impact on the occurrence of the EPs and allows them to appear in optically uniaxial materials.Furthermore, instead of eight EPs in bulk crystals, trajectories of EPs in the momentum space occur in semi-infinite crystals.In the case of finite crystals, due to the presence of two parallel interfaces, the eigenmodes are determined by a superposition of forward and backward propagating modes and thus only four EPs occur.Furthermore, in contrast to the bulk and semiinfinite crystals, there exists a discrete energy distribution of the waves, which appear at the EPs and the symmetry of these points is independent of the crystal structure.
As the two eigenmodes at an EP are degenerate in their polarization state and propagation properties, an arbitrarily polarized wave cannot be described by a superposition of the two eigenmodes as is typically described in textbooks.As a consequence, the propagation of an arbitrarily polarized wave along an EP cannot be described by this method.Furthermore, the reflection and transmission coefficients at an interface cannot be correctly calculated by means of plane waves when waves at an EP are involved.To overcome this drawback, we have presented here two approaches based on a spatially dependent amplitude of the plane wave or on a propagation matrix to describe the propagation of an arbitrarily polarized wave at an EP.We have further shown how these approaches can be used in order to compute the transmission and reflection coefficients at an interface without exhibiting a singularity if EPs are involved.

Appendices
Appendix A: Eigenvalues and Eigenvectors of a 2 Â 2 Matrix In the following, we consider a 2 Â 2 matrix given by In the trivial case that b ¼ c ¼ 0, the eigenvectors λ are λ 1 ¼ a and λ 2 ¼ d and the corresponding eigenvectors are v Here, the two eigenvectors are orthogonal even in the case that the two eigenvalues are degenerate, i.e., λ 1 ¼ λ 2 .Thus, two distinct eigenvectors are obtained, which are orthogonal to each other, for b ¼ c ¼ 0. Note, for a symmetric real-valued matrix A, there exists a coordinate transformation (rotation) such a way that the off-diagonal elements vanish.Thus, the corresponding eigenvectors of a symmetric real valued matrix are always orthogonal to each other.
For the nontrivial case, i.e., b 6 ¼ 0, the two eigenvalues are given by Note, in contrast to the case where b ¼ c ¼ 0, the eigenvectors are not necessarily orthogonal to each other anymore.A degeneration of the eigenvalues of the matrix A is then obtained for which is accompanied by a simultaneous degeneration of the eigenvectors.Excluding the trivial case mentioned above with b ¼ c ¼ 0, Equation (A2) cannot be fulfilled for real-valued matrix components and thus requires a complex-valued matrix A.
Using Equation (A2), the degenerate eigenvalues and eigenvectors are For the optical systems considered here, excluding optical activity, matrix A is symmetric, i.e., b ¼ c, and thus the eigenvalues and eigenvectors are given by Note, in the systems discussed here, the eigenvector represents the polarization of the eigenmode, and thus it is evident from Equation (A5b) that at an EP the eigenmodes are either left or right circularly polarized.

Appendix B: Films on Substrates
As discussed in Section 4, the energy of the modes depends on the phase difference of the complex refractive index at the interfaces of the substrate and the ambient material (cf., Equation (53a)).For transparent materials, the phase of the complex refractive index is 0 or π, depending on the polarization.For isotropic materials, it is well known that the phase of the complex reflection coefficient for the p-polarization exhibits a jump from 0 to π at the Brewster angle, or vice versa depending on the definition.If the two surrounding materials of the films are equal, then this jump in the phase occurs at the same in-plane wave vector for both interfaces.However, if the two surrounding materials differ from each other, the jump in the phase does not occur at the same in-plane wave vector and thus Φ ¼ ϕ s þ ϕ a ð Þ mod2π is not a monotonic function anymore and exhibits a discontinuity at the Brewster angle.As in an optically anisotropic thin film one of the two eigenmodes is strongly p-polarized (cf., Figure 19), the abovementioned consideration is still valid.Therefore, Φ as well as the mode energy of the (almost) p-polarized mode is not monotonic with respect to the in-plane wave vector and exhibits a discontinuity.In contrast to this, the phase of the complex reflection coefficient for s-polarized light is constant and thus the mode energy for the corresponding polarized mode does not exhibit a discontinuity as a function of the in-plane wave vector.Exemplarily, the mode energy and the broadening of a mode with a mode number of m ¼ 4 for a ZnO thin film with an optical axis along the x-direction are shown in Figure 27.As explained above, the p-polarized eigenmode (M 2 ) exhibits a discontinuity in its energy, whereas the energy of the s-polarized eigenmode (M 1 ) increases monotonically.In contrast to the energy, the broadening of the mode is independent of Φ (cf., Equation (53b)) and thus does not exhibit a discontinuity.Note, the discontinuity in the energy coincides with the maximum of the broadening of the mode because the change in Φ occurs at the Brewster angle.Here, also the magnitude of the corresponding reflection coefficient exhibits a minimum and thus the mode exhibits strong losses into the surrounding materials.

Appendix C: EPs in Microresonators
The properties of the EPs and their spatial distribution in microresonators are, in general, the same as for a finite crystal or a thin film.In the case of microresonators, the film is embedded between two mirrors so that the interfaces exhibit high reflectivity values, which typically cannot be achieved by surrounding the film with a single material.Microresonators are of high interest, e.g., for laser applications [4,66] and light-matter coupling, [48,67] and the presence of EPs in such systems has recently been shown. [18,19]The peculiarities caused by the high reflectivity in these systems are discussed in the following.
Typically, distributed Bragg reflectors (DBR) are used as mirrors, which yield reflectivity values of 99% and above.A typical reflectivity spectrum is shown in Figure 28.The high reflectivity of the interfaces leads to a strong suppression of mode leakage into the surrounding media and thus enhances the lifetime of the modes.Therefore, in contrast to a finite crystal or thin film, the reflectivity spectra of microresonators exhibit sharp modes in the high reflectivity region of the Bragg reflector, the so-called Bragg stop band.For sufficiently small layer thickness, i.e., d f ≈ λ 0 =2n eff , with λ 0 being the central wavelength of the Bragg stop band, the modes of two consecutive mode numbers are well separated.Although the dependence of the mode energy and broadening on the mode number, layer thickness and reflectivity of the interfaces are similar to those of finite crystals and films (Equation ( 53)), there are some deviations, which have an impact on the distribution of the EP in momentum space.
In contrast to an interface that is between two materials only, the phase of the reflection coefficient of a Bragg reflector in the vicinity of the center of the Bragg stop band depends linearly on the photon energy.The reflection coefficient can be then written as [52,53] r ¼ re iℒ EÀE s ð Þ =ℏc (C1) with r being the magnitude of the reflection coefficient, L the effective length of the DBR, and E s the central energy of the Bragg stop band.The first two quantities depend on the number layer pairs, whereas the central energy of the Bragg stop band is mainly determined by the thickness of the involved layers.An approximation for these quantities can be found in refs.[52,53]  Within this approximation, the energy and broadening of the eigenmodes are given by [52,53] The indices i ¼ 1, 2 denote the eigenmode and the properties of the top and bottom DBR are represented by the indices t and b, respectively.Note, in contrast to finite crystals and films, the denominator in Equation (C2a) depends not only on the thickness on the cavity but also on the effective length of the Bragg reflector (L i ).As typically L i ≫ d i , [53] the impact of the cavity thickness of the properties of the modes and thus on the EPs is reduced.The energy and the broadening of the eigenmode at the EP are shown in Figure 29a,b as a function of the reflectivity of the DBR.Similar to the case for a thin film, the broadening of the mode at the EP strongly decreases with the increasing reflectivity of the interface.However, whereas for the thin film the corresponding energy of the mode does not depend on the reflectivity, the energy changes slightly with increasing reflectivity of the interfaces and saturates for large reflectivity values.Although the energy does not directly depend on the reflectivity, the change is determined by the change of the effective length of the Bragg reflector (L), which depends almost linearly on the reflectivity, i.e., L ∝ R. Taking this into account, the observed behavior is well reproduced.In the case of the broadening, besides the strong reduction in the broadening caused by the reflectivity, the presence of the Bragg reflector leads to a further reduction of about one order of magnitude compared to a finite crystal or film, which is caused by the effective length of the Bragg reflector.This can be seen in Figure 29b, where the red solid and dashed line represent the calculated broadening for a film embedded between DBRs (Equation (C2b)) and a finite crystal with reflectivity values of the interfaces similar to those of the Bragg reflector (Equation ( 53)), respectively.
Besides the broadening, the energetic splitting between two eigenmodes is also reduced in a microresonator compared to that of a film.However, the reduction of the intramodal free spectral range is much smaller for large reflectivity than for the broadening.For microresonators, the intramodal finesse can be approximated by For high reflectivity values of the DBR (R ≈ 100%), the intramodal finesse reaches values larger than one (Figure 30) and thus makes a determination of the change of the eigenvalues by an encircling of the EP possible.The experimental observation of an EP in such microresonator structures was first realized by Richter et al. in 2019 in a ZnO-based microresonator [18] and 2 years later in a lead-halide perovskite-based microresonator. [19]) (b) (c) (d) Note, in the experiments performed on the ZnO-based microreonator, the reflectivity, i.e., the number of layer pairs, of the bottom DBR was smaller than what would be expected from Figure 30.The differences can be explained by the effect that here an isotropic DBR is discussed, whereas in ref. [18] an optically anisotropic DBR was used.In the latter case, the energetic mode splitting is enhanced compared to an isotropic DBR.The presence of the Bragg reflectors at the interfaces also has an impact on the k-space distribution of the EPs.In contrast to an almost straight line as observed for a finite crystal or thin film, the EPs for different reflectivity values describe a curve in k-space and saturate for large reflectivity values to a certain value.As the effective length of the DBR, which has an impact on the energy of the eigenmode, depends on the reflectivity, both the iso-energy (ΔE ¼ 0) and iso-broadening (Δγ ¼ 0) lines in the momentum space change simultaneously.Thus, the resulting crossing point between these two iso-lines for different reflectivity values is not a straight line in momentum space as for finite crystals and films.For large reflectivity values, the effective length of the DBR saturates and therefore the energy of the eigenmodes and the broadening of the mode also saturate.Furthermore, for large reflectivity values of the DBR, the difference in the reflection coefficients for two eigenmodes at the interfaces decreases.Thus, the iso-energy and iso-broadening lines in the momentum space do not differ strongly for reflectivity R ≈ 1.This holds also for their crossing point and subsequently the k-vector of the EP remains almost unchanged for large reflectivity values (cf., Figure 29).

Figure 1 .Figure 2 .
Figure 1.a-c) Difference of the complex refractive index of the two eigenmodes, i.e., jΔñj ¼ jñ 1 À ñ2 j, as a function of the polar (θ) and azimuth (ϕ) angle in a false color plot for a) KTP at E ¼ 4.6 eV, b) β-Ga 2 O 3 at E ¼ 6.3 eV, and c) K 2 Cr 2 O 7 at E ¼ 3.75 eV.The corresponding direction is defined by r ¼ cos ϕ sin θ, sin ϕ sin θ, cos θ ð Þ T .d-f ) The Stokes vector component S 3 , describing the circular polarization for the materials shown in (a-c), i.e., d) KTP, e) β-Ga 2 O 3 , and f ) K 2 Cr 2 O 7 .At the EP or rather along a singular axis, it holds S 3 ¼ AE1.Note, the circular component for each direction is the same for the two eigenmodes and only the linear components differ from each other.

Figure 3 .
Figure 3.The (a,e) real and (b,f ) imaginary part of the averaged complex refractive of the singular optic axes, i.e., ñ ¼< n > þi < k >¼ P 4 j¼1 ñj =4 for a,b) triclinic K 2 Cr 2 O 7 and e,f ) monoclinic β-Ga 2 O 3 .The difference of the c,g) real and d,h) imaginary part of the refractive index of the singular optic axes compared to the mean value of the complex refractive index of the axes c,d) triclinic K 2 Cr 2 O 7 and g,h) monoclinic β-Ga 2 O 3 .The color code of the axis is the same as in Figure 2e.The vertical dashed lines correspond to the symmetry points shown in Figure 2.

Figure 4 .
Figure 4. a) The difference of the complex refractive index (jñj ¼ jñ 1 À ñ2 j) as a function of the polar (θ) and azimuth (ϕ) angle for β-Ga 2 O 3 for an energy of E ¼ 6.5 eV.b) Schematic of the encirculation of an EP.c,d) The change of refractive index (c) and extinction coefficient (d) for a clockwise encircling of an EP (black) and an area which does not contain an EP (blue).The corresponding paths indicated in (a) by the black and blue solid lines show areas which contain and do not contain an EP, respectively.

Figure 5 .
Figure 5.The discriminant of the determinant of Det M with respect to kz for a) KTP (orthorhombic) at E ¼ 5.2 eV, b) β-Ga 2 O 3 (monoclinic) at E ¼ 5.2 eV, and c) K 2 Cr 2 O 7 at E ¼ 3.6 eV.In all cases, the in-plane component of the wave vector was set to kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p .The orientations where the forward ( kz > 0) and backward ( kz < 0) propagating modes are degenerate are marked by a white and black circle, respectively.

Figure 6 .
Figure 6.a) The Euler angles ϕ and θ for the crystal orientation having a degenerate inhomogeneous plane wave with kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p ¼ sin π=3 as a function of the Euler angle ψ for a triclinic crystal (K 2 Cr 2 O 7 ) at an energy of E ¼ 3.6 eV.The solid and dashed light represents directions where the eigenmode is almost right-or left-handed circularly polarized, respectively.The black and blue lines (first and third lines) correspond to a degeneration of the forward propagating wave ( kz > 0), whereas the gray and light blue lines (second and fourth lines) correspond to backward propagating waves ( kz < 0).The set of Euler angles which follows the symmetry relation given by Equation (28) are arranged in a 2 Â 2 block.b,c) Stereographic projection of the direction of the real (filled squares) and imaginary (empty squares) part of the wave vector (a) and of the energy flux (b) in the crystallographic system for a triclinic crystal (K 2 Cr 2 O 7 ) at an energy of E ¼ 3.6 eV.The direction of the wave vector and of the energy flux which belongs to the same wave are shown in the same color code.The inset in (a) and (b) shows an enlargement of the spatial distribution of the wave vector and energy flux.

Figure 7 .
Figure 7.The a) real and b) imaginary part of the z-component of the wave vector ( kz ) along the cone for the degeneration of the eigenvalue and eigenvector.The corresponding wave vector component for the forward (k z > 0) and backward (k z < 0) waves are shown as solid and dashed lines, respectively.

Figure 10 .
Figure 10.The same as in Figure 6 and 9 but for an orthorhombic crystal (KTP, E ¼ 3.6 eV and kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p ). a) The Euler angles φ and θ for the crystal orientation having a degenerate inhomogeneous plane wave with kx ¼ ffiffiffiffiffiffiffiffi ffi 0.75 p ¼ sinπ=3 as a function of the Euler angle ψ. b,c) Stereographic projection of the direction of the real (filled squares) and imaginary (empty squares) part of the wave vector (b) and of the energy flux (c) in the crystallographic system.The circle (dashed line) in (b) represents the orientation of the real (imaginary) part of the wave vector. 0

Figure 11 .
Figure 11.a,b) The absolute difference of the wave vector for the two eigenmodes (Δ k ¼ j k1 À k2 j) in ZnO as a function of the orientation of the ZnO crystal, represented by the two Euler angles ψ and θ for kx ¼ ffiffiffiffiffiffiffi ffi 3=4 p , which would correspond to an angle of incidence from vacuum of 60 ∘ in the a) transparent spectral range (E ¼ 2.0 eV) and b) in the absorption region (E ¼ 4.5 eV).c,d) Circular polarization component of the Stokes vector (S 3 ) of the forward c) and backward d) propagating mode in the crystal system.Note, the circular component is the same for the two forward or rather backward propagating eigenmodes. kx

Figure 12 .Figure 13 .
Figure 12.Dispersion of Euler angles a,b) ψ for the forward (a, sgn k z > 0) and backward (b, sgn k z < 0) propagating waves as well as the corresponding Euler angle c) θ for a degeneration of the wave vector of the ordinary and extraordinary wave for selected in-plane wave vectors.The solid and dashed lines represent the eigenmodes almost right (S 3 ≈ þ1) and left (S 3 ≈ À1) circularly polarized, respectively.Note, for a better presentation, only the orientations with θ < π=2 are shown.The other orientations are given in Table5.

Figure 14 .Figure 15 .Figure 16 .
Figure 14.The a) in-plane wave vector kx and b) the Euler angle ψ as a function of the photon energy for ZnO for selected Euler angles θ.The gray highlighted area in (a) represents in-plane wave vectors which can be achieved by a coupling from a vacuum state.

Figure 18 .
Figure 18.The difference of the two eigenmodes with mode number m = 7 of a ZnO thin film surrounded by vacuum in their a) energy (ΔE c ), b) broadening (Δγ), and c) complex energy (Δ Ẽ) as a function of the in-plane wave vector.The optical axis was set to be parallel to the kx direction.

Figure 19 .
Figure 19.The polarization of the eigenmodes (m = 7) of a ZnO thin film surrounded by vacuum expressed by the Stokes vector S i as a function of the in-plane wave vector.

Figure 20 .
Figure 20.Change of the a) energy and b) broadening for an encircling of the EP shown in Figure 18.The corresponding in-plane wave vector is given by kk ¼ kEP k þ k 0 cos ϕ, sin ϕ ð Þwith kEP k the in-plane wave vector at the EP and k 0 ¼ 0.1.

Figure 21 .
Figure 21.The a) energy (black circles) and b) broadening of the eigenmode with a mode number of m = 3 at the EP as a function of the thickness for a ZnO film.The energetic separation of two consecutive modes is shown as blue circles in (a).The red solid line represents the behavior calculated by Equation(53) for the energy and broadening as well as by Equation (55) for mode splitting.

Figure 22 .
Figure 22.The a) energy and b) broadening of the eigenmode with a mode number of m = 3 at the EP in 1.1 μm thick ZnO crystal as a function of the refractive index of the underlying material (substrate).The red solid line represents the behavior calculated by Equation(53).Due to the small in-plane wave vector of the mode at the EP, the reflectivity in Equation (53b) was calculated for simplicity for normal incidence assuming an isotropic refractive index of the crystal of n eff = 2.The vertical gray dashed line represents the refractive index of the ZnO crystal.The discontinuity occurs at n eff = n s , i.e., the reflection coefficient changes its sign and thus Φ changes from 0 to π.The change in the energy corresponds to πℏc=2d f n eff cos θ eff ≈ 150 meV.

Figure 23 .
Figure 23.The a) energy and b) broadening of the mode at the EP as a function of the mode number for selected refractive indices of the surrounding material for a ZnO film.c) The magnitude of the in-plane wave vector component as well as d) the position of the EP in the momentum space.The optical axis was set to be parallel to the interface and along the x-direction.For simplicity, the refractive index was set to be constant for the whole spectral range.The refractive index of the underlying material, which corresponds in the case of a film to the substrate, is denoted as n s whereas the refractive index top material was set to vacuum (n a ¼ 1.0).The red solid lines in (a) and (b) represent the calculated behavior by means of Equation (53), assuming n eff and k k to be independent of the mode number, whereas in the case of the red dashed line in (a), the change of k k was taken into account.

Figure 24 .
Figure24.The intramodal finesse ℱ m as a function a) of the mode number of 1.1 μm thick ZnO crystal for different refractive indices of the underlying material (substrate) and b) of the refractive index of the underlying material (substrate).The red line represents the value calculated by means of Equation (58a).Due to the small wave vector, the reflectivity at normal incidence was considered and an isotropic refractive index of the cavity was considered for the reflectivity, i.e., n eff ¼ 2. The refractive index of the ambient material in (a) and (b) was set to that of vacuum (n ¼ 1).The vertical gray dashed line in (b) denotes the refractive index of the crystal.

.∞ l¼0 1 l
As the propagation matrix can be expressed by a power series, i.e., e X ¼ P !X l with X ¼ ik 0 k 2Â2 z z, it can be easily seen that 1) the eigenstates of k2Â2 z correspond to the eigenstates of the wave equation and 2) the eigenvalues of k2Â2 z are the z-component of the wave vector of the corresponding eigenmode.Thus, the physical reasonable solution for the forward (backward) traveling waves of k2Â2 z is given where the real and imaginary parts of both eigenvalues of M are positive (negative).The matrix k2Â2 z describes only the change of the x-and y-component of the electric field.If we now take into account Equation 0 and thus the corresponding wave equation depends only quadratically on k2Â2 z or rather kD,2Â2 z .The solution of the wave equation yields that the matrix k2Â2 z ( kD,2Â2 z

Figure 26 .
Figure 26.a) Difference of the complex mode energies of the two eigenmodes in KTP and b) the degree of circular polarization (S 3 ) as a function of the in-plane wave vector components kx and ky for an energy of about E ≈ 2.5 eV.

Figure 27 .
Figure 27.Dispersion of the a) mode energy (E) and b) broadening (γ) of the Fabry-Perot modes with mode number m ¼ 4 in a model ZnO thin film as a function of the in-plane wave vector kx .

Figure 28 .
Figure 28.a) Schema of the microresonator structure.b) Calculated reflectivity spectrum (unpolarized) for a microresonator made of a λ=2 ZnO cavity embedded between two Bragg reflectors made of 10.5 layer-pairs of yttria stabilized zirconia (YSZ) and aluminium oxide at normal incidence.The optical axis of the ZnO cavity was set to be parallel to the surface plane.

Figure 29 .
Figure 29.The a) energy, b) broadeningand c) magnitude of the in-plane wavevector of the mode at the EP for a microresonator made of an optically uniaxial cavity layer as a function of the reflectivity of the surrounding DBRs.The optical constants of the cavity were similar to those of the discussed film in Section 4, i.e., ffiffiffiffiffi ε ⊥ p ¼ 1.991 and ffiffiffiffi ffi ε k p ¼ 2.011, and the optical axis was parallel to the k x -direction.The thickness of the cavity was set to 184 nm.The refractive indices of the DBR materials were set to n 1 ¼ 2.17 and n 2 ¼ 1.718 and the thickness of each layer fulfill n i d i ¼ 155 nm.The solid and red dashed lines in (a) and (b) represents the behaviour calculated energy and broadening for a microresonator (Equation (C2)) and for a finite crystal/film (Equation (53)), respectively.Note, the dependence of the effective length of the Bragg reflector was approximated by L ∝ R. d) The position in the k-space of the corresponding wave vector at the EP as a function of the reflectivity of the DBR.
2 and zero elsewhere, i.e., only elements of the block matrix defined by the first two lines and rows are nonzero.It should be noted that due to D z 0 ¼ 0 the third column, i.e., kD,3Â3

Table 7 .
Comparison of the properties of the EPs and of the corresponding electromagnetic wave in bulk, semi-infinite, and finite bulk crystals.