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Oblique frozen modes in periodic layered media

We study the classical scattering problem of a plane electromagnetic wave incident on the surface of semi-infinite periodic stratified media incorporating anisotropic dielectric layers with special oblique orientation of the anisotropy axes. We demonstrate that an obliquely incident light, upon entering the periodic slab, gets converted into an abnormal grazing mode with huge amplitude and zero normal component of the group velocity. This mode cannot be represented as a superposition of extended and evanescent contributions. Instead, it is related to a general (non-Bloch) Floquet eigenmode with the amplitude diverging linearly with the distance from the slab boundary. Remarkably, the slab reflectivity in such a situation can be very low, which means an almost 100% conversion of the incident light into the axially frozen mode with the electromagnetic energy density exceeding that of the incident wave by several orders of magnitude. The effect can be realized at any desirable frequency, including optical and UV frequency range. The only essential physical requirement is the presence of dielectric layers with proper oblique orientation of the anisotropy axes. Some practical aspects of this phenomenon are considered.


I. INTRODUCTION
Electromagnetic properties of periodic stratified media have been a subject of extensive research for decades (see, for example, [1,2,3] and references therein). Of particular interest has been the case of periodic stacks (1D photonic crystals) made up of lossless dielectric components with different refractive indices. Photonic crystals with one-dimensional periodicity had been widely used in optics long before the term "photonic crystals" was invented.
Let us look at the classical problem of a plane electromagnetic wave incident on the surface of semi-infinite plane-parallel periodic array, as shown in Fig. 1 extensive discussion on the subject and numerous references can be found in [4,5,6,7,8,9,10,11,12,13]. All the above effects can occur even in the simplest case of a semi-infinite periodic array of two isotropic dielectric materials with different refractive indices, for example, glass and air. The majority of known photonic crystals fall into this category.
The introduction of dielectric anisotropy, however, can bring qualitatively new features to electromagnetic properties of periodic stratified media and open up new opportunities for practical applications (see, for example, a recent publication [14]). One of such phenomena is the subject of this work. group velocity becomes vanishingly small, while the amplitude of the AFM can exceed the amplitude of the incident plane wave by several orders of magnitude.
The group velocity u of the AFM is parallel to the semi-infinite slab boundary and, therefore, the magnitude of the tangential component S T . (2) In line with Eq. (1), in the AFM regime the transmittance τ remains significant and can be even close to unity, as shown in an example in Fig. 4(a). In other words, the incident plane wave enters the slab with little reflectance, where it turns into an abnormal AFM with infinitesimally small normal component of the group velocity, huge amplitude, and huge tangential component of the energy density flux. By contrast, in the vicinity of a photonic band edge (at frequencies near ω = ω b in Fig. 4(a)), the transmittance of semi-infinite slab always vanishes, along with the normal component u z of the wave group velocity. It turns out that at a given frequency ω 0 the AFM regime can occur only for a special direction n 0 of the incident plane wave propagation This special direction of incidence always makes an oblique angle with the normal z to the layers. To find n 0 for a given ω 0 or, conversely, to find ω 0 for a given n 0 , one has to solve the Maxwell equations in the periodic stratified medium. This problem will be addressed in Section 3. In Section 2 we consider the relation between the AFM regime and the singularity of the electromagnetic dispersion relation responsible for such a peculiar behavior. If the frequency ω and the direction of incidence n do not match explicitly as prescribed by Eq.
B. The vicinity of the AFM regime Let Ψ T (z) be the transmitted electromagnetic field inside the semi-infinite slab (the explicit definition of Ψ T (z) is given in Eqs. (38) and (88)). It turns out that in the vicinity of the AFM regime, Ψ T (z) is a superposition of the extended and evanescent Bloch eigenmodes where Ψ ex (z) is an extended mode with u z > 0, and Ψ ev (z) is an evanescent mode with ℑk z > 0. As shown in an example in Fig. 5, both the contributions to Ψ T (z) have huge and nearly equal and opposite values near the slab boundary, so that their superposition (4) at z = 0 is small enough to satisfy the boundary condition (90). As the distance z from the slab boundary increases, the evanescent component Ψ ev (z) decays exponentially, while the amplitude of the extended component Ψ ex (z) remains constant and huge. As the result, the field amplitude |Ψ T (z)| 2 reaches its huge saturation value |Ψ ex | 2 at a certain distance from the slab boundary (see Eqs. (99), (100) and (101)).
When the direction of incidence n tends to its critical value n 0 for a given frequency ω 0 , the respective saturation value |Ψ ex | 2 of the AFM amplitude |Ψ T (z)| 2 diverges as | n − n 0 | −2/3 .
Conversely, when the frequency ω tends to its critical value ω 0 for a given direction of incidence n 0 , the saturation value of the AFM amplitude diverges as |ω − ω 0 | −2/3 . In the real situation, of course, the AFM amplitude will be limited by such physical factors as: (i) nonlinear effects, (ii) electromagnetic losses, (iii) structural imperfections of the periodic array, (iv) finiteness of the slab dimensions, (v) deviation of the incident radiation from a perfect plane monochromatic wave. regime. Consider a wide monochromatic beam of frequency ω incident on the surface of semiinfinite photonic slab. The direction of incidence n 0 S I is chosen so that the condition (3) of the AFM regime is satisfied at ω = ω 0 . As frequency ω tends to ω 0 from either direction, the normal component u z of the transmitted wave group velocity approaches zero, while the tangential component u ⊥ remains finite This relation together with the equality involving the refraction angle θ T , yield Hence, in the vicinity of the AFM regime, the transmitted (refracted) electromagnetic wave can be viewed as a grazing mode. The most important and unique feature of this grazing mode directly relates to the fact that the transmittance τ of the semi-infinite slab remains wave is not shown. θ I and θ T are the incidence and refraction angles, S I and S T are the energy density fluxes of the incident and transmitted waves. Both the energy density and the energy density flux in the transmitted wave are much larger than the respective values in the incident wave. However, the total power transmitted by the refracted wave is smaller by factor τ , due to much smaller cross-section area of the nearly grazing transmitted wave.
finite even at ω = ω 0 (see, for example, Fig. 4(a)). Indeed, let A I and A T be the cross-section areas of the incident and transmitted (refracted) beams, respectively. Obliviously, Let us also introduce the quantities where S I and S T are the energy density fluxes of the incident and transmitted waves. U I and U T are the total power transmitted by the incident and transmitted (refracted) beams, respectively. The expressions (8) and (9) imply that which is nothing more than a manifestation of the energy conservation law. Finally, Eq.
(10), together with the formula (7), yield where we have taken into account that τ S I cos θ I is limited (of the order of magnitude of unity) as ω → ω 0 . By contrast, in the vicinity of the photonic band edge the transmittance τ of the semi-infinite slab vanishes along with the energy density flux S T of the transmitted (refracted) wave.
The expressions (7) and (11) show that in the vicinity of the AFM regime, the transmitted wave behaves like a grazing mode with huge and nearly tangential energy density flux S T and very small (compared to that of the incident beam) cross-section area A T , so that the total power U T = A T S T associated with the transmitted wave cannot exceed the total power U I of the incident wave: U T = τ U I ≤ U I .
The above qualitative consideration is only valid on the scales exceeding the size L of the unit cell (which is of the order of magnitude of c/ω) and more importantly, exceeding the transitional distance l = (ℑk ev ) −1 from the slab boundary where the evanescent mode contribution to the resulting electromagnetic field Ψ T (z) is still significant. The latter means that the width of both the incident and the refracted beams must be much larger than l. If the above condition is not met, we cannot treat the transmitted wave as a beam, and the expressions (7) through (11) do not apply. Instead, we would have to use the explicit electrodynamic expressions for Ψ T (z), such as the asymptotic formula (101). Note that if the direction n of the incident wave propagation and the frequency ω exactly match In some respects, the remarkable behavior of the AFM, is similar to that of the frozen mode related to the phenomenon of electromagnetic unidirectionality in nonreciprocal magnetic photonic crystals [15,16]. In a unidirectional photonic crystal, electromagnetic radiation of a certain frequency ω 0 can propagate with finite group velocity u z only in one of the two opposite directions, say, from right to left. The problem with the electromagnetic unidirectionality, though, is that it essentially requires the presence of magnetic materials with strong circular birefringence (Faraday rotation) and low losses at the frequency range of interest. Such materials are readily available at the microwave frequencies, but at the infrared and optical frequency ranges, finding appropriate magnetic materials is highly problematic.
Thus, at frequencies above 10 12 Hz, the electromagnetic unidirectionality along with the respective nonreciprocal magnetic mechanism of the frozen mode formation may prove to be impractical. By contrast, the occurrence of AFM does not require the presence of magnetic or any other essentially dispersive components in the periodic stack. Therefore, the AFM regime can be realized at any frequencies, including the infrared, optical, and even ultraviolet frequency ranges. The only essential physical requirement is the presence of anisotropic dielectric layers with proper orientation of the anisotropy axes. An example of such an array is shown in Fig. 2.
In Section 2 we establish the relation between the phenomenon of AFM and the electro- Finally, in Section 5 we discuss some practical aspects of the phenomenon.

II. DISPERSION RELATION WITH THE AFM
Now we establish the connection between the phenomenon of AFM and the electromagnetic dispersion relation ω k , k = (k x , k y , k z ) of the periodic stratified medium. In a plane-parallel stratified slab, the tangential components (k x , k y ) of the Bloch wave vector k always coincide with those of the incident plane wave in Figs. 1, 3, and 6 while the normal component k z is different from that of the incident wave. To avoid confusion, in further consideration, the z component of the Bloch wave vector k inside the periodic slab will be denoted as k without the subscript z, namely Inside periodic stack: k = (k x , k y , k).
The value of k is found by solving the Maxwell equations in the periodic stratified medium for given ω and (k x , k y ); k is defined up to a multiple of 2π/L, where L is the period of the layered structure.
Consider now the frequency ω as function of k for fixed (k x , k y ). A typical example of such a dependence is shown in Fig. 7(a). A large gap at the lowest frequencies is determined by the value of the fixed tangential components (k x , k y ) of the quasimomentum k. This gap vanishes in the case of normal incidence, when k x = k y = 0. An alternative and more convenient representation for the dispersion relation is presented in Fig. 7 The pair of values (n x , n y ) coincide with the tangential components of the unit vector n defining the direction of the incident plane wave propagation. The dependence ω(k) for fixed (n x , n y ) or for fixed (k x , k y ) will be referred to as the axial dispersion relation.
Suppose that for k = k 0 and ω = ω 0 = ω k 0 , one of the spectral branches ω (k) develops a stationary inflection point for given (k x , k y ) = (k 0x , k 0y ), i.e., ∂ω ∂k kx,ky k= k 0 = 0; The value in Eq. (13) is the axial component of the group velocity, which vanishes at k = k 0 . Observe that u x = ∂ω ∂k x k,ky and u y = ∂ω ∂k y k,kx , representing the tangential components of the group velocity, may not be zeros at k = k 0 .
Notice that instead of (13), one can use another definition of the stationary inflection point ∂ω ∂k nx,ny k= k 0 = 0, The partial derivatives in Eqs. (16) are taken at constant (n x , n y ), rather than at constant (k x , k y ). Observe that the definitions (13) and (16) are equivalent, and we will use both of them.
In Fig. 4(b) we reproduced an enlarged fragment of the upper spectral branch of the axial dispersion relation in Fig. 7(b). For the chosen (n x , n y ), this branch develops a stationary inflection point (16) at ω = ω 0 and k = k 0 . The extended Bloch eigenmode with ω = ω 0 and k = k 0 , associated with the stationary inflection point, turns out to be directly related to the axially frozen mode (AFM).
In Sections 3 and 4, based on the Maxwell equations, we prove that the singularity (16) (or, equivalently, (13)) indeed leads to the very distinct AFM regime in the semi-infinite periodic stack. We also show that a necessary condition for such a singularity and, therefore, a necessary condition for the AFM existence is the following property of the axial dispersion relation of the periodic stack This property will be referred to as the axial spectral asymmetry. Evidently, the axial dispersion relations presented in Fig. 7, satisfy this criterion. Leaving the proof of the above statements to Section 3, let us look at the constraints imposed by the criterion (17) on the geometry and composition of the periodic stack.

A. Conditions for the axial spectral asymmetry
First of all, notice that a periodic array would definitely have an axially symmetric dispersion relation if the symmetry group G of the periodic stratified medium includes any of the following two symmetry operations where m z is the mirror plane parallel to the layers, 2 z is the 2-fold rotation about the z axis, and R is the time reversal operation. Indeed, since 2 z (k x , k y , k) = (−k x , −k y , k) and which implies the relation (18) for arbitrary (k x , k y ). The same is true for the mirror plane Consequently, a necessary condition for the axial spectral asymmetry (17) of a periodic stack is the absence of the symmetry operations (19), i.e., In reciprocal (nonmagnetic) media, where by definition, R ∈ G, instead of Eq. (20) one can use the following requirement Note, that the axial spectral symmetry (18) is different from the bulk spectral symmetry For example, the space inversion I and/or the time reversal R, if present in G, ensure the bulk spectral symmetry (22), but neither I nor R ensures the axial spectral symmetry (18). (21) to deferent periodic stacks.

Application of the criterion
The condition (21) for the axial spectral asymmetry imposes certain restrictions on the geometry and composition of the periodic stratified medium, as well as on the direction of the incident wave propagation.
a. Restrictions on the geometry and composition of periodic stack. First of all, observe that a common periodic stack made up of isotropic dielectric components with different refractive indices, always has axially symmetric dispersion relation (18), no matter how complicated the periodic array is or how many different isotropic materials are involved. To prove this, it suffices to note that such a stack always supports the symmetry operation 2 z .
In fact, the symmetry operation 2 z holds in the more general case when all the layers are either isotropic, or have a purely in-plane anisotropŷ Obviously, the in-plane anisotropy (23) does not remove the symmetry operation 2 z and, therefore, the property (18) of the axial spectral symmetry holds in this case. Thus, we can state that in order to display the axial spectral asymmetry, the periodic stack must include at least one anisotropic component, either uniaxial or biaxial. In addition, one of the principle axes of the respective dielectric permittivity tensorε must make an oblique angle with the normal to the layers, which means that at least one of the two components ε xz and ε yz of the respective dielectric tensor must be nonzero.
The above requirement gives us a simple and useful idea on what kind of periodic stratified media can support the axial spectral asymmetry and the AFM regime. But this is not a substitute for the stronger symmetry criterion (20) or (21). For example, although the periodic stack in Fig. 8 includes the A layers identical to those in not meet the criterion (20) for the axial spectral asymmetry. Indeed, the stack in Fig. 8 supports the mirror plane m z , which, according to the expression (19), ensures the axial spectral symmetry.
b. Restriction on the direction of incident wave propagation Consider now an important particular case k x = k y = 0 of the normal incidence. The criterion (17) reduces now to the simple requirement of the bulk spectral asymmetry, which is prohibited in nonmagnetic photonic crystals due to the time reversal symmetry. Therefore, in the nonmagnetic case, we have the following additional condition for the axial spectral asymmetry implying that the AFM cannot be excited in a nonmagnetic semi-infinite stack by a normally incident plane wave, i.e., the incident angle must be oblique.
Conditions (21) and (25) may not be necessary in the case of nonreciprocal magnetic stacks (see the details in [15]). But as we mentioned earlier, at frequencies above 10 12 Hz, the nonreciprocal effects in common nonconducting materials are negligible. Therefore, in order to have a robust AFM regime in the infrared or optical frequency range, we must satisfy both requirements (21) and (25), regardless of whether or not nonreciprocal magnetic materials are involved.
As soon as the above conditions are met, one can always achieve the AFM regime at any desirable frequency ω within certain frequency range ∆ω. The frequency range ∆ω is determined by the stack geometry and the dielectric materials used, while a specific value of ω within the range can be selected by the direction n of the light incidence.

B. Periodic stack with two layers in unit cell
The simplest and the most practical example of a periodic stack supporting the axial spectral asymmetry (17) and, thereby, the AFM regime, is shown in Fig. 2. It is made up of anisotropic A layers alternated with isotropic B layers. The respective dielectric permittivity tensors areε For simplicity, we assumeμ The stack in Fig. 2 has the monoclinic symmetry 2 y /m y with the mirror plane m y normal to the y -axis. Such a symmetry is compatible with the necessary condition (21) for the AFM existence. But as we will see below, the symmetry (28) imposes additional constraints on the direction n of the incident wave propagation.
In Fig. 9 we show the axial dispersion relation ω (k) of this periodic array, computed for four different directions (n x , n y ) of incident wave propagation. These four cases cover all the possibilities, different in terms of symmetry.
In the case (a) of normal incidence, when n x = n y = 0, the dispersion relation is axially symmetric, as must be the case with any reciprocal periodic stratified medium (see the explanation after Eq. (24)).
In the case (b), when n x = 0 and n y = 0, the two necessary conditions (21) and (25) for the axial spectral asymmetry are met. Yet, those conditions prove not to be sufficient.
Indeed, if n x = 0, either of the symmetry operations 2 y and m ′ y ≡ m y × R imposes the relation which implies the axial spectral symmetry. Neither stationary inflection point, nor AFM can occur in this case.
In the case (c), when n x = 0 and n y = 0, the situation is more complicated. The quasimomentum k lies now in the x − z plane, which coincides with the mirror plane m y .
Therefore, every Bloch eigenmode Ψ k (z) can be classified as a pure TE or pure TM mode, depending on the Ψ k (z) parity with respect to the mirror reflection m y for TE mode m y Ψ k (z) = −Ψ k (z) ; for TM mode: The TE modes have axially symmetric dispersion relation for TE modes: Indeed, the component ε xz of the dielectric tensorε A does not affect the TE modes, because in this case the electric component E(r, t) of the electromagnetic field is parallel to the y axis. As a consequence, the axial dispersion relation of the TE spectral branches is similar to that of the isotropic case with ε xz = 0, where it is always symmetric. By contrast, for the TM modes we have E(r, t) ⊥ y. Therefore, the TM modes are affected by ε xz and display axially asymmetric dispersion relation as seen in Fig. 9(c). We wish to remark, though, that the equality (32) cannot be derived For instance, if we impose the equality n x = n y and change the incident angle only, it turns out that every single spectral branch at some point develops a stationary inflection point (16) and, thereby, displays the AFM at the respective frequency. If we want the AFM at a specified frequency ω 0 , then we will have to adjust both n x and n y .

A. Reduced Maxwell equations
We start with the classical Maxwell equations for time-harmonic fields in nonconducting where In a lossless dielectric medium, the material tensorsε ( r) andμ ( r) are Hermitian. In a stratified medium, the tensorsε ( r) andμ ( r) depend on a single Cartesian coordinate z, and the Maxwell equations (34) can be recast as Solutions for Eq. (36) are sought in the following form The substitution (37) transforms the system of six linear equation (36) into a system of four linear differential equations The explicit expression for the Maxwell operator M (z) is where where (n x , n y ) are defined in Eq. (12).
Notice that in the case of normal incidence, the Maxwell operator is drastically simplified This is the case we dealt with in [16] when considering the phenomenon of electromagnetic unidirectionality in nonreciprocal magnetic photonic crystals. By contrast, the objective of  [17,18,19], and references therein). For more detailed studies of J -Hermitian and J -unitary operators see [20].

B. The transfer matrix
The Cauchy problem for the reduced Maxwell equation (38) has a unique solution where the 4 × 4 matrix T (z, z 0 ) is so-called transfer matrix. From the definition (45) of the transfer matrix it follows that The matrix T (z, z 0 ) is uniquely defined by the following Cauchy problem The equation (47), together with J -Hermitivity (42) of the Maxwell operator M (z), imply that the matrix T (z, z 0 ) is Junitarity, i.e., (see the proof in Appendix 1). The J -unitarity (48) of the transfer matrix imposes strong constraints on its eigenvalues (see Eq. (61)). It also implies that The transfer matrix T S of a stack of layers is a sequential product of the transfer matrices Additional information related to the transfer matrix formalism can be found in [17,18,19] and references therein.
The definition (45) of the T -matrix together with Eq. (54) give Introducing the transfer matrix of a primitive cell we have from Eq. (55) Thus, the eigenvectors of the transfer matrix T L of the unit cell are uniquely related to the eigenmodes of the reduced Maxwell equation (38) through the relations The respective four eigenvalues of T L are the roots of the characteristic equation The J-unitarity (48) of T L imposes the following restriction on the eigenvalues (59) for any given ω and (k x , k y ) In view of the relation (61), one has to consider three different situation. The first possibility relates to the case of all four Bloch eigenmodes being extended. The second possibility relates to the case of two extended and two evanescent modes. The last possibility relates the case of a frequency gap, when all four Bloch eigenmodes are evanescent.
Observe that the relation valid in the case of normal incidence (see Refs. [15,16]), may not apply now.

Axial spectral symmetry
Assume that the transfer matrix T L is similar to its inverse where U is an invertible 4 × 4 matrix. This assumption together with the property (48) of J-unitarity, imply the similarity of T L and T † This relation imposes additional restrictions on the eigenvalues (59) for a given frequency ω and given (k x , k y ) The relation (67) is referred to as the axial spectral symmetry, because in terms of the corresponding axial dispersion relation, it implies the equality (18) for every spectral branch.
If the sufficient condition (65) for the axial spectral symmetry is not in place, then we can have for a given ω and (k x , k y ) which implies the axial spectral asymmetry (17).

D. Stationary inflection point
The coefficients of the characteristic polynomial F (X) in Eq. (60) are functions of ω and (k x , k y ). Let F 0 (X) be the characteristic polynomial at the stationary inflection point (16), where ω = ω 0 and (k x , k y ) = (k 0x , k 0y ). The stationary inflection point (16) can also be defined as follows This relation requires the respective value of X 0 = exp (ik 0 L) to be a triple root of the characteristic polynomial F 0 (X) implying A small deviation of the frequency ω from its critical value ω 0 changes the coefficients of the characteristic polynomial and removes the triple degeneracy of the solution X 0 or, in terms of the axial quasimomentum k where The three solutions (72) can also be rearranged as The real k ex in (74) relates to the extended mode Ψ ex (z), with u z = 0 at ω = ω 0 . The other two solutions, k ev and k EV = k * ev , correspond to a pair of evanescent modes Ψ ev (z) and Ψ EV (z) with positive and negative infinitesimally small imaginary parts, respectively.
Those modes are truly evanescent (i.e., have ℑk = 0) only if ω = ω 0 , but it does not mean that at ω = ω 0 , the eigenmodes Ψ ev (z) and Ψ EV (z) become extended. In what follows we will take a closer look at this problem.

Eigenmodes at the frequency of AFM
Consider the vicinity of stationary inflection point (13). As long as ω = ω 0 , the four eigenvectors (58) of the transfer matrix T L comprise two extended and two evanescent Bloch solutions. One of the extended modes (say, Φ k 1 ) corresponds to the non-degenerate real root X 1 = e ik 1 L of the characteristic equation (60). This mode has negative axial group velocity u z (k 1 ) < 0 and, therefor, is of no interest for us. The other three eigenvectors of T L correspond to three nearly degenerate roots (71). As ω approaches ω 0 , these three eigenvalues become degenerate, while the respective three eigenvectors Φ k 2 , Φ k 3 , and Φ k 4 become collinear The latter important feature relates to the fact that at ω = ω 0 , the matrix T L has a nontrivial Jordan canonical form and, therefore, cannot be diagonalized. It is shown rigorously in [16], that the very fact that the T L eigenvalues display the singularity (71), implies that at ω = ω 0 , the matrix T L has the canonical form (76). In line with (75), the matrix T L from Eq. (76) has only two (not four!) eigenvectors: 1. Φ k 1 = Ψ k 1 (0), corresponding to the non-degenerate root X 1 and relating to the extended mode with u z < 0; 2. Φ k 0 = Ψ k 0 (0), corresponding to the triple root X 0 and related to the AFM.
The other two solutions of the Maxwell equation (38) at ω = ω 0 are general Floquet eigenmodes, which do not reduce to the canonical Bloch form (54). Yet, they can be related to Ψ k 0 (z). Indeed, following the standard procedure (see, for example, [21,22]), consider an extended Bloch solution Ψ k (z) of the reduced Maxwell equation (38) where both operators M (z) and L (z) are functions of ω and (k x , k y ). Assume now that the axial dispersion relation ω (k) has a stationary inflection point (13) at k = k 0 . Differentiating Eq. (77) with respect to k at constant (k x , k y ) gives, with consideration for Eq. (13), This implies that at k = k 0 , both functions are also eigenmodes of the reduced Maxwell equation at ω = ω 0 . Representing Ψ k (z) in the and substituting Eq. (79) into (78) we get whereΨ k 0 (z) = (∂ k ψ k (z)) k=k 0 e ik 0 z andΨ ′ k 0 (z) = ∂ 2 kk ψ k (z) k=k 0 e ik 0 z are auxiliary Bloch functions (not eigenmodes).
To summarize, at the frequency ω 0 of AFM, there are four solutions for the reduced Maxwell equation (38) The first two solutions from (82) are extended Bloch eigenmodes with u z < 0 and u z = 0, respectively. The other two solutions diverges as the first and the second power of z, respectively, they are referred to as general (non-Bloch) Floquet modes.
Deviation of the frequency ω from ω 0 removes the triple degeneracy (76) of the matrix T L , as seen from Eq. (71). The modified matrix T L can now be reduced to a diagonal form with the set (58) of four eigenvectors comprising two extended and two evanescent Bloch solutions.

E. Symmetry considerations
In Section 2, we discussed the relation between the symmetry of the axial dispersion relation of a periodic stack, and the phenomenon of AFM. At this point we can prove that indeed, the axial spectral asymmetry (17) is a necessary condition for the occurrence of the stationary inflection point and for the AFM associated with such a point. As we have seen earlier in this Section, the stationary inflection point relates to a triple root of the characteristic polynomial F (X) from Eq. (60). Since F (X) is a polynomial of the fourth degree, it cannot have a symmetric pair of triple roots, that would have been the case for axially symmetric dispersion relation. Hence, only asymmetric axial dispersion relation ω (k) can display a stationary inflection point (13) or, equivalently, (16), as shown in Fig. 4(b). In this respect, the situation with the AFM is somewhat similar to that of the frozen mode in unidirectional magnetic photonic crystals [16]. The difference lies in the physical nature of the phenomenon. The bulk spectral asymmetry (24) leading to the effect of electromagnetic unidirectionality, essentially requires the presence of nonreciprocal magnetic materials. By contrast, the axial spectral asymmetry (17) along with the AFM regime can be realized in perfectly reciprocal periodic dielectric stacks with symmetric bulk dispersion relation (22). On the other hand, the axial spectral asymmetry essentially requires an oblique light incidence, which is not needed for the bulk spectral asymmetry.
Another important symmetry consideration is that in the vicinity of the stationary inflection point (13), all four Bloch eigenmodes (58) must have the same symmetry, which means that all of them must belong to the same one-dimensional irreducible representation of the Bloch wave vector group. This condition is certainly met when the direction defined by (n x , n y ) is not special in terms of symmetry.
Let us see what happens if the above condition is not in place. Consider the situation (31), when at any given frequency ω and fixed (n x , n y ) = (n x , 0), two of the Bloch eigenmodes are TE modes and the other two are TM modes. Note, that TE and TM modes belong to different one-dimensional representations of the Bloch wave vector group. In such a case, the transfer matrix T L can be reduced to the block-diagonal form The respective characteristic polynomial F (X) degenerates into where F T E (X) and F T M (X) are independent second degree polynomials describing the TE and TM spectral branches, respectively. Obviously, in such a situation, the transfer matrix cannot have the nontrivial canonical form Eq. (76), and the respective axial dispersion relation cannot develop a stationary inflection point (69), regardless of whether or not the axial spectral asymmetry is in place.

A. Boundary conditions
In vacuum (to the left of semi-infinite slab in Fig. 1) the electromagnetic field Ψ V (z) is a superposition of the incident and reflected waves At the slab boundary we have where The transmitted wave Ψ T (z) inside the semi-infinite slab is a superposition of two Bloch (in the case of a finite slab, all four eigenmodes (58) would contribute to Ψ T (z)). The eigenmodes Ψ 1 (z) and Ψ 2 (z) can be both extended (with u x > 0), one extended and one evanescent (with u x > 0 and ℑk > 0, respectively), or both evanescent (with ℑk > 0), depending on which of the three cases (62), (63), or (64) we are dealing with. In particular, in the vicinity of the AFM (e.g., the vicinity of ω 0 in Fig. 4(b)), we always have the situation (63). Therefore, in the vicinity of AFM, Ψ T (z) is a superposition of the extended eigenmode Ψ ex (z) with the group velocity u z > 0, and the evanescent mode Ψ ev (z) with ℑk > 0 The asymptotic expressions for the respective wave vectors k ex and k ev in the vicinity of AFM are given in Eq. (74).
When the frequency ω exactly coincides with the frequency ω 0 of the AFM, the representation (88) for Ψ T (z) is not valid. In such a case, according to Eq. (74), there is no evanescent modes at all. It turns out that at ω = ω 0 , the electromagnetic field inside the slab is a superposition of the extended mode Ψ k 0 (z) and the (non-Bloch) Floquet eigenmode Since the extended eigenmode Ψ k 0 (z) has zero axial group velocity u z , it does not contribute to the axial energy flux S z . By contrast, the divergent non-Bloch contribution Ψ 01 (z) is associated with the finite axial energy flux S z > 0, although the notion of group velocity does not apply here. The detailed analysis is carried out in the next subsection.
Knowing the eigenmodes inside the slab and using the standard electromagnetic boundary one can express the amplitude and composition of the transmitted wave Ψ T and reflected wave Ψ R , in terms of the amplitude and polarization of the incident wave Ψ I . This gives us the transmittance and reflectance coefficients (2) of the semi-infinite slab, as well as the electromagnetic field distribution Ψ T (z) inside the slab, as functions of the incident wave polarization, the direction n of incidence, and the frequency ω.

B. Field amplitude inside semi-infinite slab
In what follows we assume that ω can be arbitrarily close but not equal to ω 0 , unless otherwise is explicitly stated. This will allow us to treat the transmitted wave Ψ T (z) as a superposition (88) of one extended and one evanescent mode. Since evanescent modes do not transfer energy in the z direction, the extended mode is solely responsible for the axial According to Eq. (109), S z does not depend on z and can be expressed in terms of the semi-infinite slab transmittance τ from Eq. (2) where S I = S I z is the axial energy flux of the incident wave, which is set to be unity.
The energy density W ex associated with the extended mode Ψ ex (z) can be expressed in terms of the axial component u z of its group velocity and the axial component S z (Ψ ex ) of the respective energy density flux In close proximity of the AFM frequency ω 0 , we have according to Eq. (13) where ω ′′′ 0 is defined in Eq. (73). Differentiating Eq. (94) with respect to k ∂ω ∂k kx,ky and plugging Eq. (95) into (93) yields where the transmittance τ depends on the incident wave polarization, the frequency ω, and the direction of incidence (n x , n y ) = (ck x /ω, ck y /ω). Formula (96) implies that the energy density W ex and, therefore, the amplitude |Ψ ex (z)| = |Φ ex | of the extended mode inside the stack diverge in the vicinity of the AFM regime The divergence of the extended mode amplitude |Φ ex | imposes the similar behavior on the amplitude |Ψ ev (0)| = |Φ ev | of the evanescent mode at the slab boundary. Indeed, the boundary condition (90) requires that the resulting field Φ T = Φ ex + Φ ev remains limited to match the sum Φ I + Φ R of the incident and reflected waves. The relation (90) together with (97) imply that there is a destructive interference of the extended Φ ex and evanescent Φ ev modes at the stack boundary Here Φ k 0 is the normalized eigenvector of T L in Eq. (76); K is a dimensionless parameter.
The expression (98) is in compliance with the earlier made statement (75) that the columnvectors Φ ex and Φ ev become collinear as ω → ω 0 .

Space distribution of electromagnetic field in the AFM regime
The amplitude |Ψ ex (z)| of the extended Bloch eigenmode remains constant and equal to |Φ ex | from (97), while the amplitude of the evanescent contribution to the resulting field decays as At z ≫ (ℑk ev ) −1 , the destructive interference (98) of the extended and evanescent modes becomes ineffective, and the only remaining contribution to Ψ T (z) is the extended mode Ψ ex (z) with huge and independent of z amplitude (97). This situation is graphically demonstrated in Fig. 5.
Let us see what happens when the frequency ω tends to its critical value ω 0 . According to Eqs. (57) and (88), at z = NL, N = 0, 1, 2, ..., the resulting field Ψ T (z) inside the slab can be represented as Substituting k ex and k ev from Eq. (74) in (100), and taking into account the asymptotic relation (98), we have Although this asymptotic formula is valid only for z = NL, N = 0, 1, 2, ..., it is obviously consistent with the expression (80) for the non-Bloch solution Ψ 10 (z) of the Maxwell equation (38) at ω = ω 0 .

The role of the incident wave polarization
The incident wave polarization affects the relative contributions of the extended and evanescent components to the resulting field Ψ T (z) in Eq. (88). In addition, it also affects the overall transmittance (2). The situation here is similar to that of the normal incidence, considered in [16]. There are two special cases, merging into a single one as ω → ω 0 .
The first one occurs when the elliptic polarization of the incident wave is chosen so that it produces a single extended eigenmode Ψ ex (z) inside the slab (no evanescent contribution to Ψ T (z)). In this case, Ψ T (z) reduces to Ψ ex (z), and its amplitude |Ψ T (z)| remains limited and independent of z. As ω approaches ω 0 , the respective transmittance τ vanishes in this case, and there is no AFM regime. The second special case is when the elliptic polarization of the incident wave is chosen so that it produces a single evanescent eigenmode Ψ ev (z) inside the slab (no extended contribution to Ψ T (z)). In such a case, Ψ T (z) reduces to Ψ ev (z), and the amplitude |Ψ T (z)| decays exponentially with z in accordance with Eq. (99). The respective transmittance τ in this latter case is zero regardless of the frequency ω, because evanescent modes do not transfer energy. Importantly, as ω approaches ω 0 , the polarizations of the incident wave that produce either a sole extended or a sole evanescent mode become indistinguishable, in accordance with Eq. (75). In the vicinity of the AFM regime, the maximal transmittance τ is achieved for the incident wave polarization orthogonal to that exciting a single extended or evanescent eigenmode inside the semi-infinite stack.
So far we have been focusing on the axial electromagnetic field distribution, as well as the axial energy flux S z inside semi-infinite slab. At the same time, in and near the AFM regime, the overwhelmingly stronger energy flux occurs in the tangential direction. Let us take a closer look at this problem.
The axial energy flux S z is exclusively provided by the extended contribution Ψ ex (z) to the resulting field Ψ T (z), because the evanescent mode Ψ ev (z) does not contribute to S z .
Neither |Ψ ex | nor S z depends on z (see Eq. (109)). By contrast, both the extended and the evanescent modes determine the tangential energy flux S τ (z). Besides, according to Eq.
(109), the tangential energy flux depends on z. Far from the AFM regime, the role of the evanescent mode is insignificant, because Ψ ev (z) is appreciable only in a narrow region close to the slab boundary. But the situation appears quite different near the AFM frequency.
Indeed, according to Eq. (99), the imaginary part of the respective Bloch wave vector k ev becomes infinitesimally small near the critical point. As a consequence, the evanescent mode extends deep inside the slab, so does its role in formation of S τ (z). The tangential energy flux S τ (z) as function of z can be directly obtained using formula (111) and the explicit expression for Ψ T (z) = Ψ ex (z) + Ψ ev (z). Although the explicit expression for S τ (z) is rather complicated and cumbersome, it has very simple and transparent structure. Indeed, the tangential energy flux can be represented in the following form where the tangential group velocity u τ behaves regularly at ω = ω 0 . Therefore, the magnitude and the space distribution of the tangential energy flux S τ (z) in and near the AFM regime literally coincides with that of the electromagnetic energy density W (z), which is proportional to |Ψ T (z)| 2 . A typical picture of that is shown in Fig. 5(a).

V. SUMMARY
As we have seen in the previous Section, a distinctive characteristic of the AFM regime is that the incident monochromatic radiation turns into a very unusual grazing wave inside the slab, as shown schematically in Figs. 3 and 6. Such a grazing wave is significantly different from that occurring in the vicinity of the total internal reflection regime, where the transmitted wave also propagates along the interface. The most obvious difference is that near the regime of total internal reflection, the reflectivity approaches unity, which implies that the intensity of the transmitted (refracted) wave vanishes. By contrast, in the case of AFM the light reflection from the interface can be small, as shown in an example in Fig.   4(a). Thus, in the AFM case, a significant portion of the incident light gets converted into the grazing wave (the AFM) with huge amplitude, compared to that of the incident wave.
For this reason, the AFM regime can be of great utility in many applications.
Another distinctive feature of the AFM regime relates to the field distribution inside the periodic medium. The electromagnetic field of the AFM can be approximated by a divergent Floquet eigenmode Ψ 10 (z) from (80), whose magnitude |Ψ 10 (z)| 2 increases as z 2 , until nonlinear effects or other limiting factors come into play. In fact, the field amplitude inside the slab can exceed the amplitude of the incident plane wave by several orders of magnitude, depending on the quality of the periodic array, the actual number of the layers, and the width of the incident light beam.
Looking at the z component of light group velocity and energy flux, we see a dramatic slowdown of light in the vicinity of the AFM regime, with all possible practical applications extensively discussed in the literature (see, for example, [9,10,11,12], and references therein). In principle, there can be a situation when the tangential components (u x , u y ) of the group velocity also vanish in the AFM regime, along with the axial component u z . Although we did not try to achieve such a situation in our numerical experiments, it is not prohibited and might occur if the physical parameters of the periodic array are chosen properly. In such a case, the AFM regime reduces to its particular case -the frozen mode regime with u = 0 inside the periodic medium. This regime would be similar to that considered in [16], with one important difference: it is not related to the magnetic unidirectionality and, hence, there is no need to incorporate nonreciprocal magnetic layers in the periodic array. The latter circumstance allows to realize the frozen mode regime at the infrared, optical, and even UV frequency range.
To prove it, notice that Eq. (102) implies Now, let us find the respective Cauchy problem for T −1 (z). Since which is identical to that for T † (z) from Eq. (104). Since both Cauchy problems (104) and (106) have unique solutions, their similarity implies the relation (103) of J-unitarity.