Electromagnetic unidirectionality in magnetic photonic ... - UCI Math

PHYSICAL REVIEW B 67, 165210 共2003兲

Electromagnetic unidirectionality in magnetic photonic crystals A. Figotin and I. Vitebskiy Department of Mathematics, University of California at Irvine, California 92697 共Received 22 August 2002; revised manuscript received 22 November 2002; published 28 April 2003兲 We study the effect of electromagnetic unidirectionality, which can occur in magnetic photonic crystals under certain conditions. A unidirectional periodic medium, being perfectly transparent for an electromagnetic wave of certain frequency, ‘‘freezes’’ the radiation of the same frequency propagating in the opposite direction. One of the most remarkable manifestations of the unidirectionality is that while the incident radiation can enter the unidirectional slab in either direction with little or even no reflectance, it cannot escape from there getting trapped inside the periodic medium in the form of the coherent frozen mode. Having entered the slab, the wave slows down dramatically and its amplitude increases enormously, creating unique conditions for nonlinear phenomena. Such a behavior is an extreme manifestation of the spectral nonreciprocity, which can only occur in gyrotropic photonic crystals. Unidirectional photonic crystals can be made of common ferro- or ferrimagnetic materials alternated with anisotropic dielectric components. A key requirement for the property of unidirectionality is the proper spatial arrangement of the constitutive components. DOI: 10.1103/PhysRevB.67.165210

PACS number共s兲: 78.20.Bh, 41.20.Jb, 78.20.Ls, 42.65.⫺k

I. INTRODUCTION

k⫽k z ,

A. Unidirectional photonic crystals

Photonic crystals are spatially periodic composites made up of lossless dielectric components. As a consequence of spatial periodicity, the electromagnetic frequency spectrum of a photonic crystal develops a band-gap structure similar to that of electrons in semiconductors and metals 共see, for instance, Refs. 1– 6 and references therein兲. Gyrotropic photonic crystals are those in which at least one of the constitutive components is a magnetic material 共a ferromagnet or a ferrite兲 displaying the Faraday rotation.7–9 Such materials are often referred to as gyrotropic or bigyrotropic. If a gyrotropic photonic crystal satisfies certain symmetry conditions formulated in Ref. 10, its bulk electromagnetic dispersion relation ␻ (kជ ) may display asymmetry with respect to the Bloch wave vector kជ ,

␻ 共 kជ 兲 ⫽ ␻ 共 ⫺kជ 兲 ,

共1兲

as shown in Fig. 1. The bulk spectral asymmetry 共1兲 by no means occurs automatically in any magnetic photonic crystal. Quite the opposite, only special periodic arrays of magnetic and other dielectric components can produce the effect.25 An example of a such periodic stack is shown in Fig. 2. The degree of the spectral asymmetry depends on the magnitude of circular birefringence of the gyrotropic component, as well as on some other geometric and physical parameters of the periodic array. Detailed theoretical analysis of the problem along with a number of specific examples are provided in Ref. 10. The property of bulk spectral asymmetry has various physical consequences, one of which is the effect of unidirectional wave propagation. Let us consider a transverse monochromatic wave propagating along a symmetry direction z of a gyrotropic photonic crystal. The Bloch wave vector kជ , as well as the group velocity uជ (kជ )⫽ ⳵␻ (kជ )/ ⳵ kជ are parallel to z. Let us denote 0163-1829/2003/67共16兲/165210共20兲/$20.00

u 共 k 兲 ⫽ ⳵␻ 共 k 兲 / ⳵ k,

共2兲

and suppose that one of the spectral branches ␻ (k) has a stationary inflection point at k⫽k 0 , ␻ ⫽ ␻ 0 ,

⳵␻ ⳵k

冏

⫽0; k⫽k 0

⳵ 2␻ ⳵k2

冏

⫽0; k⫽k 0

⳵ 3␻ ⳵k3

冏

⫽0,

共3兲

k⫽k 0

as shown in Fig. 1. Note that there are two propagating 共extended兲 Bloch waves with frequency ␻ ⫽ ␻ 0 : one with k ⫽k 0 , and the other with k⫽k 1 . Obviously, only one of the two waves can transfer electromagnetic energy—the one with k⫽k 1 and the group velocity u(k 1 )⬍0. The Bloch eigenmode with k⫽k 0 has zero group velocity u(k 0 )⫽0 and does not transfer energy. This latter eigenmode associated with stationary inflection point 共3兲 is referred to as the frozen mode. As one can see in Fig. 1, none of the eigenmodes with ␻ ⫽ ␻ 0 has positive group velocity and therefore none of the electromagnetic eigenmodes can transfer the energy from left to right at this particular frequency! Thus a photonic crystal with the dispersion relation similar to that in Fig. 1 displays the property of electromagnetic unidirectionality at ␻ ⫽ ␻ 0 . Such a remarkable effect can be viewed as an extreme manifestation of the spectral asymmetry 共1兲. According to Ref. 10, the effect of unidirectionality can occur in magnetic photonic crystals made up of common dielectric and ferro- or ferrimagnetic components 共at least at frequencies below 1012Hz). There are two key physical requirements for that: 共i兲 The space arrangement of the constitutive components must satisfy certain symmetry criterion for spectral asymmetry. This criterion, specified in Ref. 10, rules out all nonmagnetic and the majority of magnetic photonic crystals. The space arrangement of magnetic and other constitutive components must be complex enough to allow for the bulk spectral asymmetry 共1兲. 共ii兲 The magnetic constituent 共for instance, ferrite兲 must display significant circular birefringence at frequency range of interest, for example, several percent or more. Failure to

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©2003 The American Physical Society


A. FIGOTIN AND I. VITEBSKIY

FIG. 1. An example of asymmetric bulk electromagnetic dispersion relation of a periodic magnetic stack. At ␻ ⫽ ␻ 0 , k⫽k 0 , one of the spectral branches develops a stationary inflection point 共3兲 associated with the frozen mode. ␻ b is the edge of the lowest frequency band. The graphs 共a兲 and 共b兲 represent two different choices of the Brillouin zone. The values ␻ and k are expressed in units of c/L and 1/L, respectively.

satisfy this condition does not formally rule out the phenomenon of unidirectionality, but it makes the magnitude of the effect insignificant. Indeed, weak Faraday rotation leads to a small value of the third derivative ( ⳵ 3 ␻ / ⳵ k 3 ) k⫽k 0 in Eq. 共3兲, which, in turn, pushes the stationary inflection point ␻ 0 in Fig. 1 too close to the photonic band edge ␻ b . The simplest and, perhaps, the most practical periodic structures displaying the property of unidirectionality are one-dimensional 共1D兲 periodic magnetic stacks, an example of which is presented in Fig. 2. If the above two conditions

are met, one can always achieve electromagnetic unidirectionality at a given frequency ␻ 0 by adjusting at least two different physical and/or geometrical parameters, such as • the ratio ␳ ⫽F/A of the layers thicknesses, • the misalignment angle ␸ ⫽ ␸ 1 ⫺ ␸ 2 between anisotropic dielectric layers, • magnetic permeability and/or electric permittivity of the layers 共this can be done by application of external homogeneous magnetic or electric field19,20兲. Physical manifestations of the electromagnetic unidirectionality prove to be rather universal and dependent solely on the dispersion relation ␻ (kជ ) in the vicinity of the frozen mode frequency ␻ 0 . An essential characteristic determining the magnitude of the respective electromagnetic abnormalities is the dimensionless parameter

␾⫽

FIG. 2. A simplest periodic magnetic stack supporting asymmetric bulk dispersion relation. Each primitive cell L⫽2A⫹F comprises three layers: two anisotropic dielectric layers 1 and 2 of thickness A and with misaligned in-plain anisotropy, and one magnetic layer of thickness F and magnetization shown by the arrows. The misalignment angle between adjacent layers 1 and 2 must be different from 0 and ␲ /2.

1

冉冊 ⳵ 3␻

␻ 0L 3 ⳵ k 3

,

共4兲

k⫽k 0

where L is the length of the primitive cell of the photonic crystal. For instance, periodic stacks made of different constitutive materials and having completely different geometry, display, nevertheless, very similar behavior in the vicinity of the frozen mode frequency, provided that they have comparable values of the parameter ␾ from Eq. 共4兲. For this reason, all numerical examples considered in this paper are based on a single magnetic periodic stack shown in Fig. 2 and described in detail in Appendix A. These examples illustrate the universal features of electromagnetic behavior of all unidirectional photonic crystals.

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ELECTROMAGNETIC UNIDIRECTIONALITY IN . . .

FIG. 3. Asymmetric dispersion relations of the periodic structures slightly modified compared to that related to Fig. 1. In both cases, the stationary inflection point of Fig. 1 evolves into a simple inflection point with ␻ ⬙kk (k)⫽0: 共a兲 the ratio ␳ ⫽F/A of the layers thicknesses exceeds the the critical value ␳ 0 by a third; 共b兲 ␳ ⬍ ␳ 0 by a third. Here ␳ 0 is the ‘‘unidirectional’’ ratio corresponding to the situation in Fig. 1.

If any of the physical or geometrical parameters of a unidirectional stack is altered, the stationary inflection point 共3兲 can turn into a regular inflection point corresponding to a finite group velocity, as shown in Fig. 3共a兲, or a pair of close inflection points, as in the situation in Fig. 3共b兲. This blurs and weakens the effects associated with electromagnetic unidirectionality. At first sight, the existence of the frozen mode related to a stationary inflection point 共3兲 does not require the spectral asymmetry 共1兲. Indeed, a hypothetical symmetric dispersion relation in Fig. 4共a兲 would have a pair of stationary inflection points, although there would be no spectral asymmetry, let alone unidirectionality, in such a case. But in fact, the situation similar to that in Fig. 4共a兲 cannot occur regardless of the complexity of the composite structure. At any given frequency, there cannot be more than one stationary inflection point 共3兲 in the electromagnetic spectrum. Therefore the fro-

zen mode cannot exist in periodic stacks, either magnetic or nonmagnetic, with symmetric dispersion relations ␻ (kជ ) ⫽ ␻ (⫺kជ ). On the other hand, whenever the electromagnetic dispersion relation has a stationary inflection point 共3兲共i.e., the frozen mode兲, it always displays the property of unidirectionality at the same frequency. This implies that a hypothetical dispersion relation in Fig. 4共b兲 having a stationary inflection point at ␻ ⫽ ␻ 0 but not displaying electromagnetic unidirectionality at the respective frequency, cannot occur either. Recall that the unidirectionality means the existence of propagating modes with only negative 共or only positive兲 group velocity at a given frequency and given direction of propagation. The above statements suggest a strict one-to-one correspondence between the existence of the frozen mode and the property of unidirectionality at the same frequency in peri-

FIG. 4. Hypothetical dispersion relations, which cannot exist in any periodic stack regardless of its complexity: 共a兲 symmetric dispersion relation with two stationary inflection points at the same frequency ␻ 0 ; 共b兲 asymmetric dispersion relation with stationary inflection point 共the frozen mode兲 at ␻ ⫽ ␻ 0 , but without the property of unidirectionality at the same frequency.

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FIG. 5. Forward 共left-to-right兲 normal incidence on the surface of a semi-infinite unidirectional slab with the dispersion relation in Fig. 1; the frequency ␻ lies within the range ␻ a ⬍ ␻ ⬍ ␻ b . ⌿ I (z) and ⌿ R (z) are the incident and reflected waves in vacuum; ⌿ ex (z) and ⌿ e v (z) are the extended 共with u⬎0) and the evanescent 共with Im k⬎0) contributions to the transmitted wave ⌿ T (z).

odic layered media. This is a consequence of the fact that the dispersion relation ␻ (k) of a periodic stack is determined by characteristic equation of the fourth degree. Indeed, at a given frequency ␻ , there must be a total of four real and complex solutions for the wave vector kជ 储 z. Taking into consideration that a stationary inflection point 共3兲 is always associated with a triple real root of the characteristic equation, one can come to the following conclusions: 共i兲 There cannot be more than one frozen mode at the same frequency ␻ 0 . For instance, the dispersion relation in Fig. 4共a兲 showing a couple of inflection points at the same frequency cannot occur. 共ii兲 There must be one and only one additional eigenmode at the frequency ␻ 0 of the frozen mode with the wave vector k different from k 0 . For instance, the dispersion relation in Fig. 4共b兲 showing three additional eigenmodes at the frequency of the frozen mode, cannot occur either. Detailed consideration of the bulk electromagnetic spectra in infinite periodic gyrotropic stacks is presented in Sec. II, where we thoroughly analyze a peculiar behavior of the extended and evanescent modes in nonreciprocal periodic stacks. Emphasis is given to the vicinity of stationary inflection point where the phenomenon of unidirectionality occurs. Importantly, if the frequency ␻ exactly coincides with ␻ 0 from Eq. 共3兲, the electromagnetic field inside the unidirectional periodic array does not reduce to a linear superposition of canonical Bloch eigenmodes. The latter peculiarity is related to the triple degeneracy of the stationary inflection point. The results of Sec. II are further applied to semiinfinite and finite unidirectional slabs.

FIG. 6. Backward 共right-to-left兲 normal incidence on the surface of a semi-infinite unidirectional slab with the dispersion relation in Fig. 1; the frequency ␻ lies within the range ␻ a ⬍ ␻ ⬍ ␻ b . ⌿ I (z) and ⌿ R (z) are the incident and reflected waves in vacuum; ⌿ EX (z) and ⌿ EV (z) are the extended 共with u⬍0) and the evanescent 共with Im k⬍0) contributions to the transmitted wave ⌿ T (z).

with the z axis in Eqs. 共2兲 and 共3兲. Due to the spectral asymmetry of the slab, the situation of the reversed incidence presented in Fig. 6 appears to be quite different and will be considered later on. Let S I ⬎0, S R ⭐0 and S T ⭓0 be the energy flux of the incident (⌿ I ), reflected (⌿ R ) and transmitted (⌿ T ) waves, respectively. The energy conservation yields S I ⫹S R ⫽S T

共5兲

or, equivalently S T⫽ ␶ S I ,

S R ⫽⫺ ␳ S I ,

␳ ⫽1⫺ ␶ ,

where ␶ and ␳ are the normal transmittance and reflectance of the semi-infinite slab, respectively (0⭐ ␶ ⭐1,0⭐ ␳ ⭐1). Assume that the wave frequency ␻ lies within the frequency range

␻ a⬍ ␻ ⬍ ␻ b

The phenomenon of unidirectionality is associated with unique electromagnetic properties of periodic media in the vicinity of the frozen mode frequency ␻ 0 . Some preliminary conclusions can be drawn from the energy conservation consideration. Consider a plane electromagnetic wave impinging on the surface of a semi-infinite unidirectional photonic crystal, as shown in Fig. 5. The direction z of wave propagation is perpendicular to the photonic slab boundary and coincides

共7兲

in Fig. 1. In such a case, the transmitted wave ⌿ T (z) inside the slab is a superposition of one extended 共propagating兲 Bloch eigenmode ⌿ ex (z) 共the one with u⬎0) and one evanescent mode ⌿ e v (z) 共the one with Im k⬎0), as shown in Fig. 5. Evanescent eigenmodes, which are not shown in the dispersion relation in Fig. 1, do not contribute to the energy flux, therefore the extended eigenmode ⌿ ex (z) is the only one contributing to the energy flux S T transmitted inside the slab. In the case of a single propagating mode, the energy flux S T can be expressed in terms of the mode energy density W ex and its group velocity u from Eq. 共2兲 S T ⫽uW ex .

B. Electromagnetic properties of a semi-infinite unidirectional slab

共6兲

共8兲

The fact that the group velocity u vanishes as ␻ → ␻ 0 , implies two possible scenarios, depending on whether the energy flux S T inside the slab also vanishes as ␻ → ␻ 0 . The most obvious scenario would be S T →0,␶ →0, as ␻ → ␻ 0 ,

共9兲

which implies a total reflectance of the incident wave at ␻ ⫽ ␻ 0 . Such a behavior would be similar to what commonly occurs in any semi-infinite photonic slab in the vicinity of a

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FIG. 7. Transmittance ␶ e of semi-infinite unidirectional slab vs frequency ␻ 共in units of c/L) for the case of forward incidence shown in Fig. 5. The characteristic frequencies are explained in Fig. 1. The incident wave polarization ជ 储 x; 共b兲 linear, is: 共a兲 linear, with E ជ with E 储 y; 共c兲 elliptic, corresponding to maximal transmittance at ␻ ⫽ ␻ 0 ; 共d兲 elliptic, that produces a single extended mode ⌿ ex (z) inside the slab 共no evanescent mode contribution兲.

photonic band edge, where the group velocity of the extended eigenmode also vanishes. Specifically, referring to the example in Fig. 1, we have in the vicinity of the photonic band edge at ␻ ⫽ ␻ b S T →0, ␶ →0,

as ␻ → ␻ b

共10兲

ally increases until reaches its saturation value 兩 ⌿ ex 共 z 兲兩 2 ⬃ 兩 ␻ ⫺ ␻ 0 兩 ⫺2/3.

The distance l from the slab boundary, at which the wave intensity approaches its maximum value 共12兲, is also strongly dependent on 兩 ␻ ⫺ ␻ 0 兩 ,

as illustrated in Fig. 7. This common situation occurs in any semi-infinite photonic crystal near photonic band edges. By contrast, it turns out that in the vicinity of the frozen mode frequency ␻ 0 we have, instead of Eq. 共9兲, S T →S 0 ⬎0, ␶ → ␶ 0 ⬎0, W T →⬁, as ␻ → ␻ 0 .

共11兲

In such a case, the incident wave ⌿ I with the frequency ␻ close to ␻ 0 can enter the semi-infinite unidirectional slab with little or even no reflectance 关see Figs. 7共a兲,共c兲 where the transmittance ␶ remains finite in the vicinity of ␻ 0 ]. Having entered the slab, the incident wave converts into nearly frozen extended mode ⌿ ex (z) and slows down dramatically. Immediately upon entering the slab, the wave amplitude 兩 ⌿ T (z) 兩 2 is limited, as shown in Fig. 8共a兲, but then it gradu-

共12兲

l⬃

冏冉冊 ⳵ 3␻ ⳵k3

共␻⫺␻0兲 k⫽k 0

冏

⫺1

1/3

.

共13兲

Relatively small amplitude of the electromagnetic field ⌿ T (z) in the transient region zⰆl is due to a destructive interference of the nearly frozen mode ⌿ ex (z) and the evanescent mode ⌿ e v (z) with Re k⬎0. As illustrated in Figs. 8 and 9共a兲 and 共b兲, both contributions have huge and nearly equal and opposite values near the slab boundary, so that their superposition ⌿ T (0)⫽⌽ T ⫽⌽ ex ⫹⌽ e v at z⫽0 is relatively small, as shown in Fig. 9共c兲. The destructive interference allows to satisfy the boundary condition 共81兲 at the semi-infinite slab boundary. At zⰇl the contribution of the FIG. 8. Amplitude of the resulting electromagnetic field ⌿ T (z) and its extended and evanescent components ⌿ ex (z) and ⌿ e v (z) inside unidirectional slab. z is the distance from the slab surface in units of L. Frequency ␻ is close to ␻ 0 共specifically, ␻ ⫺ ␻ 0 ⫽0.05␻ 0 ). The amplitude 兩 ⌿ I 兩 2 of the incident radiation is equal to unity.

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FIG. 9. Destructive interference of the extended and the evanescent components of electromagnetic field ⌽ T ⫽⌿ T (0) at the surface of semi-infinite unidirectional slab for the case of forward incidence: 共a兲 extended 共nearly frozen兲 contribution 兩 ⌽ ex 兩 2 2 ជ ⫽ 兩 ⌿ ex (0) 兩 for E 储 y; 共b兲 evanescent contribution 兩 ⌽ ex 兩 2 ជ 储 y; 共c兲 the re⫽ 兩 ⌿ ex (0) 兩 2 for E sulting field amplitude 兩 ⌽ T 兩 2 ជ 储 y; 共d兲 the ⫽ 兩 ⌽ ex ⫹⌽ e v 兩 2 for E ជ 储 x. resulting field amplitude for E

evanescent mode ⌿ e v (z) decays exponentially, while the contribution of the extended mode ⌿ ex (z) remains constant and huge. As a result, the total electromagnetic field amplitude ⌿ T (z) inside the slab gradually increases with the distance z until reaches its maximum value of ⌿ ex (z) from Eq. 共12兲, as illustrated in Fig. 8共a兲. If the frequency ␻ exactly coincides with ␻ 0 , the transmitted wave ⌿ T (z) inside the unidirectional slab is not a superposition of canonical Bloch eigenmodes, and its amplitude inside the slab diverges at ␻ ⫽ ␻ 0 :

兩 ⌿ T 共 z 兲兩 2 ⬃z 2 ,

as z→⬁

The effect of unidirectional freezing proves to be rather robust when some physical or geometrical parameters of the original unidirectional array are slightly altered. For instance, when the relative thickness ␳ ⫽F/A of the layers in the periodic structure in Fig. 2 is increased or decreased by a third 关the respective modified dispersion relations are presented in

共14兲

as shown in Fig. 10. The phenomenon described by formulas 共12兲 and 共14兲 and illustrated in Figs. 8 and 10 can be viewed as unidirectional freezing of the incident electromagnetic wave inside the semi-infinite unidirectional slab. It is accompanied by a dramatic slowdown of the transmitted wave inside the slab, as well as a huge increase in its amplitude. Remarkably, the transmittance ␶ e of the semi-infinite slab remains finite and can be even close to 100%. By contrast, in the situation when the plane electromagnetic wave of the same frequency ␻ close or equal to ␻ 0 impinges on the surface of the same unidirectional stack but from the opposite direction, as shown in Fig. 6, nothing extraordinary occurs. The incident wave gets partially reflected, and the rest continues inside the slab in the form of the extended Bloch eigenmode ⌿ EX (z) with finite group velocity u(k 1 )⬍0 and finite amplitude 兩 ⌿ EX (z) 兩 2 ⫽ 兩 ⌽ EX 兩 2 , as illustrated in Fig. 11. Such an extreme asymmetry between the cases of forward and backward incidence justifies the term unidirectional freezing for what happens in a semi-infinite unidirectional slab.

FIG. 10. Amplitude 兩 ⌿ T (z) 兩 2 of electromagnetic field inside unidirectional slab vs the distance z 共in units of L) from the slab surface; the frequency ␻ coincides with the frozen mode frequency ␻ 0 . Polarization of the forward incident wave is linear, with Eជ 兩兩 x 关compare with Fig. 8共a兲兴.

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FIG. 11. The amplitude of electromagnetic field ⌽ T and its extended (⌽ EX ) and evanescent (⌽ EV ) components at the surface of semi-infinite unidirectional slab in Fig. 6 for the case of backward incidence: 共a兲 extended contribuជ 储 y; 共b兲 evanestion 兩 ⌽ EX 兩 2 for E cent contribution 兩 ⌽ EV 兩 2 for Eជ 储 y; 共c兲 the resulting field amplitude ជ 储 y; 共d兲兩 ⌽ T 兩 2 ⫽ 兩 ⌽ EX ⫹⌽ EV 兩 2 for E the resulting field amplitude 兩 ⌽ T 兩 2 for Eជ 储 x.

Figs. 3共a兲 and 共b兲, respectively兴, the frozen mode blurs, but the surge in electromagnetic field amplitude inside the slab remains quite significant—more than an order of magnitude. In Sec. IV, we consider the transmittance of a finite gyrotropic photonic slab, which is a finite fragment of a unidirectional photonic crystal. As long as the number of layers constituting a finite slab is small, the electromagnetic properties of the slab does not show any indication of the unidirectionality of the respective infinite or semi-infinite periodic stacks. But when the number N of the elementary fragments L in Fig. 2 is large, the finite slab does show some distinct behavior in the vicinity of the frozen mode frequency ␻ 0 . For instance, the dependence of the slab transmittance on the polarization of the incident radiation is similar to that of the semi-infinite slab. In Figs. 12共a兲 and 共b兲 one can see that for certain elliptical polarization of the forward incident radiation, the thick unidirectional slab becomes virtually transparent in the vicinity of the frozen mode frequency ␻ 0 . This particular polarization coincides with that shown in Fig. 7共c兲 and provides the maximal forward transmittance ␶ e of the respective semi-infinite slab. In addition to this, both the thick finite slab and the respective unidirectional semiinfinite stack become totally reflective in the vicinity of ␻ 0 , if the polarization of the incident wave is orthogonal to the previous one, as shown in Figs. 7共d兲 and 12共b兲, respectively. II. TRANSVERSE ELECTROMAGNETIC WAVES IN PERIODIC GYROTROPIC MEDIA: ELECTROMAGNETIC UNIDIRECTIONALITY.

We consider the basic features of extended and evanescent eigenmodes characteristic of nonreciprocal periodic arrays. Particular attention is given to the effect of unidirectionality. The results of this section are used in the following study of the electromagnetic properties of unidirectional slabs. A. Definitions and notations

Electromagnetic properties of gyrotropic layered media have been a subject of numerous publications 共see, for example Refs. 10–16, and references therein兲. Our objective here is to introduce those concepts, definitions, and notations, which are necessary for understanding the electromagnetic properties of unidirectional photonic crystals. We consider the simplest and the most important case of layered dielectric media, which supports transverse electromagnetic waves with alternating field components

ជ 共 z 兲, H ជ 共 z 兲, D ជ 共 z 兲 , Bជ 共 z 兲⬜zជ . E

共15兲

The direction z of wave propagation is normal to the layers, as shown in Fig. 5. In such a case, the time harmonic Maxwell equations i␻ i␻ ជ 共 rជ 兲 ⫽⫺ D ជ 共 rជ 兲 Bជ 共 rជ 兲 , “⫻H c c

共16兲

⳵ ⳵ i␻ i␻ ជ 共 z 兲 ⫽ Bជ 共 z 兲 , ␴ˆ H ជ 共 z 兲 ⫽⫺ D ជ 共 z 兲, E ⳵z c ⳵z c

共17兲

ជ 共 rជ 兲 ⫽ “⫻E can be recast as

This section starts with a brief discussion of bulk electromagnetic properties of gyrotropic periodic layered structures. 165210-7

␴ˆ



FIG. 12. Forward transmittance of a thick unidirectional slab with N⫽32 in the vicinity of the frozen mode frequency ␻ 0 . The elliptical polarization of the incident wave is: 共a兲 the same as in Fig. 7共c兲, that provides the maximal transmittance; 共b兲 the same as in Fig. 7共d兲, that provides total reflectance in both cases. Small deviation of the extreme points from ␻ ⫽ ␻ 0 is due to a finite thickness of the slab.

where, in accordance with Eq. 共15兲, all the fields are twodimensional vectors lying in the x-y plane and

␴ˆ ⫽

冋

0

⫺1

1

0

册

ˆ 共 z 兲⌿共 z 兲⫽␻⌿共 z 兲, M where

E x共 z 兲

. ⌿共 z 兲⫽

ជ (z) The transverse alternating electric and magnetic fields E ជ and H (z) in Eq. 共17兲 are related to the electric and magnetic ជ (z) and Bជ (z) by common linear constitutive inductions D relations ជ 共 z 兲 ⫽␧ˆ 共 z 兲 Eជ 共 z 兲 , Bជ 共 z 兲 ⫽ ␮ˆ 共 z 兲 H ជ 共 z 兲. D

冋

冋

␧ xx 共 z 兲

␧ xy 共 z 兲

␧* xy 共 z 兲

␧ y y共 z 兲

册

,

␮ˆ 共 z 兲 ⫽

冋

共18兲

␮ xx 共 z 兲

␮ xy 共 z 兲

␮* xy 共 z 兲

␮ y y共 z 兲

H x共 z 兲

H y共 z 兲

,

册

␮ˆ ⫺1 共 z 兲 ␴ˆ ⳵ . ⳵z 0

共22兲

The transfer matrix of a layered structure

册

共19兲

are frequency dependent and take different values in different layers of the stack. The substitution of Eq. 共18兲 into Eq. 共17兲 gives

␴ˆ

E y共 z 兲

0 c ˆ 共 z 兲⫽ M i ⫺␧ˆ ⫺1 共 z 兲 ␴ˆ

The Hermitian anisotropic tensors ␧ˆ 共 z 兲 ⫽

冋册

共21兲

i␻ i␻ ⳵ ⳵ ជ 共 z 兲 ⫽ ␮ˆ 共 z 兲 H ជ 共 z 兲 ; ␴ˆ H ជ 共 z 兲 ⫽⫺ ␧ˆ 共 z 兲 Eជ 共 z 兲 . E ⳵z c ⳵z c 共20兲

ជ (z) are continuous functions of z, even The fields Eជ (z) and H ˆ (z) along with D ជ (z) and Bជ (z) are not. if ␧ˆ (z) and ␮ The reduced Maxwell equations 共20兲 can also be recast in a compact form,

The transfer-matrix formalism is particularly useful in electrodynamics of layered media composed of anisotropic and/or gyrotropic layers. Below we introduce the basic definitions and notations, consistent with those of Ref. 10. More information on the subject can be found in Refs. 13–16, and references therein. The reduced time-harmonic Maxwell equations 共21兲 constitute a system of four ordinary linear differential equations of the first order. Its general solution is a linear superposition of four eigenmodes, ⌿ 共 z 兲 ⫽C 1 ⌿ 1 共 z 兲 ⫹C 2 ⌿ 2 共 z 兲 ⫹C 3 ⌿ 3 共 z 兲 ⫹C 4 ⌿ 4 共 z 兲 .

共23兲

The four coefficients C i in Eq. 共23兲 can be uniquely related to the four transverse field components 共22兲 at a given point z,

165210-8



冋册

related to the phenomenon of spectral asymmetry 共1兲. For an extended discussion see the next subsection.

C1

ˆ 共z兲 ⌿ 共 z 兲 ⫽W

C2

C3

,

共24兲

B. Extended and evanescent modes in nonreciprocal periodic stacks

C4

1. Characteristic equation

ˆ (z) is a nonsingular 4⫻4 matrix where W ˆ 共 z 兲 ⫽ 关 ⌿ 1 共 z 兲 ⌿ 2 共 z 兲 ⌿ 3 共 z 兲 ⌿ 4 共 z 兲兴 W

共25兲

composed of the column vectors ⌿ i (z) from Eq. 共23兲. The equality 共24兲 yields a one-to-one correspondence between the electromagnetic field components ⌿(z) at any two different locations z 1 and z 2 , ˆ 共 z 2 ,z 1 兲 ⌿ 共 z 1 兲 , ⌿ 共 z 2 兲 ⫽T

共26兲

where the 4⫻4 matrix ˆ 共 z 2 ,z 1 兲 ⫽W ˆ 共 z2兲W ˆ ⫺1 共 z 1 兲 T

共28兲

where Tˆ (z)⫽Tˆ ⫺1 (⫺z). In addition, in homogeneous materials without linear magnetoelectric effect, the matrix Tˆ (z) and Tˆ ⫺1 (z) are similar, Tˆ 共 z 兲 ⫽UTˆ ⫺1 共 z 兲 U ⫺1 ,

⌿ k 共 z⫹L 兲 ⫽e ikL ⌿ k 共 z 兲 ,

共29兲

⌿ k 共 z⫹L 兲 ⫽Tˆ共 z⫹L,z 兲 ⌿ k 共 z 兲 .

Tˆ L ⌽ k ⫽e ikL ⌽ k ,

We also introduce the transfer matrix of the mth homogeneous layer Tˆ m ⫽Tˆ (z m ), where z m is the layer thickness. The single-layer transfer matrix Tˆ m depends on the layer thickˆ m . The explicit exness z m and material tensors ␧ˆ m and ␮ pressions for the Tˆ matrices of anisotropic and gyrotropic layers are rather cumbersome, and those we use are presented in Appendix A. The T matrix of a stack of layers is the product of the matrices T m constituting the stack Tˆ S ⫽

兿m Tˆ m .

det共 Tˆ L ⫺ ␨ Î 兲 ⬅F 共 ␨ 兲 ⫽ ␨ 4 ⫹ P 3 ␨ 3 ⫹ P 2 ␨ 2 ⫹ P 1 ␨ ⫹1⫽0, 共38兲 where, according to Ref. 10, P 1 ⫽ P 3* , P 2 ⫽ P 2* .

共39兲

Introducing the real coefficients R⫽Re P 1 , P⫽Im P 1 ,

共40兲

we recast Eq. 共38兲 as F 共 ␨ 兲 ⫽ ␨ 4 ⫹ 共 R⫺i P 兲 ␨ 3 ⫹ P 2 ␨ 2 ⫹ 共 R⫹i P 兲 ␨ ⫹1⫽0 共41兲 or, in the more symmetrical form,

共32兲

for an arbitrary stack. At the same time, the similarity relation ⫺1 , Tˆ S ⫽UTˆ ⫺1 S U

共37兲

of Tˆ L are the roots of the characteristic equation

共31兲

Equations 共30兲 and 共31兲 imply that det Tˆ S ⫽1

共36兲

where Tˆ L ⫽Tˆ(L,0) is the T matrix of the primitive cell of the periodic stack, while ⌽ k ⫽⌿ k (0) is one of the four Bloch solutions ⌿ k (z) for the reduced Maxwell equations 共20兲 at z⫽0. Equation 共36兲 implies that the Bloch eigenvectors ⌽ k uniquely relate to those of the transfer matrix Tˆ L . The respective four eigenvalues

␨ i ⫽e ik i L , i⫽1,2,3,4 共30兲

共35兲

Comparing Eqs. 共35兲 and 共34兲 we get at z⫽0

implying that det Tˆ 共 z 兲 ⫽1.

共34兲

where L is the length of the primitive cell of the periodic stack, k is the Bloch wave vector 共2兲, and ⌿ k (z) is the respective column vector 共22兲. The quasimomentum k is defined uniquely up to a multiple of 2 ␲ /L. It follows from the definition 共26兲 of the T matrix that

共27兲

is referred to as the transfer matrix.26 In homogeneous media, the transfer matrix 共26兲 has translation symmetry ˆ 共 z 2 ,z 1 兲 , Tˆ 共 z 2 ⫺z 1 兲 ⫽T

Bloch solutions for the Maxwell equations 共20兲 in a periodic medium satisfy

M 共 ␨ 兲 ⫽ ␨ ⫺2 F 共 ␨ 兲 ⫽ ␨ 2 ⫹ 共 R⫺i P 兲 ␨ ⫹ P 2 ⫹ 共 R⫹i P 兲 ␨ ⫺1 ⫹ ␨ ⫺2 ⫽0. Plugging

␨ ⫽cos共 kL 兲 ⫹i sin共 kL 兲

共33兲

analogous to Eq. 共29兲, may not hold for some gyrotropic stacks composed of three or more layers. This is directly

共42兲

in Eq. 共42兲 yields yet another form of the characteristic equation

165210-9



M 共 k 兲 ⫽⫺2⫹ P 2 ⫹2R cos共 kL 兲 ⫹2 P sin共 kL 兲 ⫹4 cos2 共 kL 兲 ⫽0,

共43兲

relates to the case of a frequency gap, when all four Bloch eigenmodes are evanescent 共the frequency range

␻ b⬍ ␻

where all the coefficients are now real.

in Fig. 1兲. Equation 共32兲 implies that in all cases

2. Extended and evanescent solutions

The coefficients of the characteristic equation are expressed in terms of the elements of the matrix Tˆ L . Those elements are functions of the physical parameters of the constitutive layers and the frequency ␻ . For any given frequency ␻ , the characteristic equation defines a set of four values 兵 ␨ 1 , ␨ 2 , ␨ 3 , ␨ 4 其 , or equivalently, 兵 k 1 ,k 2 ,k 3 ,k 4 其 . Real k 共roots with 兩 ␨ 兩 ⫽1) correspond to propagating Bloch waves 共extended modes兲, while complex k 共roots with 兩 ␨ 兩 ⫽1) correspond to evanescent modes. Evanescent modes are relevant near photonic crystal boundaries and other structural irregularities. The characteristic equation 共42兲 implies that for any given frequency ␻ , if ␨ is a root, then 1/␨ * is also a root

共51兲

␨ 1 ␨ 2 ␨ 3 ␨ 4 ⫽1

共52兲

k 1 ⫹k 2 ⫹k 3 ⫹k 4 ⬅0.

共53兲

or, equivalently

C. Spectral symmetry vs spectral asymmetry

If all the coefficients in the characteristic equation 共38兲 are real 关that amounts to P⫽0 in Eq. 共40兲兴, then for a given frequency ␻

兵 ␨ 1 , ␨ 2 , ␨ 3 , ␨ 4 其 ⫽ 兵 ␨ *1 , ␨ 2* , ␨ *3 , ␨ 4* 其 ,

共54兲

共44兲

or, in terms of the Bloch wave vectors

共45兲

if P⫽0, then 兵 k 1 ,k 2 ,k 3 ,k 4 其 ⫽ 兵 ⫺k * 1 ,⫺k * 2 ,⫺k * 3 ,⫺k * 4 其. 共55兲

In view of the statement 共44兲, one has to consider three different situations. The first possibility,

Observe that the relation 共54兲 together with Eq. 共44兲 ensure similarity of the matrix Tˆ L and Tˆ L⫺1 ,

兩 ␨ 1 兩 ⫽ 兩 ␨ 2 兩 ⫽ 兩 ␨ 3 兩 ⫽ 兩 ␨ 4 兩 ⫽1,

if P⫽0, then Tˆ L ⫽UTˆ L⫺1 U ⫺1 .

or, equivalently, if k is a solution, then k * is also a solution.

共46兲

or, equivalently,

Conversely

k 1 ⬅k * 1 , k 2 ⬅k 2* , k 3 ⬅k 3* , k 4 ⬅k 4* , relates to the case of all four Bloch eigenmodes being extended 共see, for instance, the frequency range 0⬍ ␻ ⬍ ␻ a

共47兲

if P⫽0, then Tˆ L ⫽UTˆ L⫺1 U ⫺1 for any U. In terms of the dispersion relation ␻ (k), the relation 共55兲 together with Eq. 共45兲 imply the spectral reciprocity 共spectral symmetry兲 of the Bloch eigenmodes,

in Fig. 1兲. The second possibility,

if P⫽0 then

兩 ␨ 1 兩 ⫽ 兩 ␨ 2 兩 ⫽1; ␨ 4 ⫽1/␨ * 3 ; where 兩 ␨ 3 兩 , 兩 ␨ 4 兩 ⫽1, 共48兲

兵␻共 k 1 兲, ␻共 k 2 兲, ␻共 k 3 兲, ␻共 k 4 兲其

or, equivalently

⫽ 兵 ␻ 共 ⫺k 1 兲 , ␻ 共 ⫺k 2 兲 , ␻ 共 ⫺k 3 兲 , ␻ 共 ⫺k 4 兲其 . 共56兲

k 1 ⫽k 1* , k 2 ⫽k 2* , k 3 ⫽k * 4 , where k 3 ⫽k 3* , k 4 ⫽k 4* , relates to the case of two extended and two evanescent modes 共the frequency range

␻ a⬍ ␻ ⬍ ␻ b

共49兲

in Fig. 1兲. The last possibility,

␨ 2 ⫽1/␨ * 1 ; ␨ 4 ⫽1/␨ 3* ; where 兩 ␨ 1 兩 , 兩 ␨ 2 兩 , 兩 ␨ 3 兩 , 兩 ␨ 4 兩 ⫽1, 共50兲 or, equivalently,

In view of the symmetry consideration of Ref. 10, the relation 共56兲 holds for all nonmagnetic and for the majority of magnetic photonic crystals. The appearance of complex coefficients P i in Eq. 共38兲关that amounts to P⫽0 in Eq. 共40兲兴 leads to violation of the relation 共55兲 for a given frequency ␻ , if P⫽0 then 兵 k 1 ,k 2 ,k 3 ,k 4 其 ⫽ 兵 ⫺k * 1 ,⫺k 2* ,⫺k 3* ,⫺k 4* 其 , 共57兲 which in terms of the dispersion relation ␻ (k) implies the spectral asymmetry

k 1 ⫽k * 2 , k 3 ⫽k * 4 , where k 1 ⫽k 1* , k 2 ⫽k * 2 , k 3 ⫽k * 3 ,k 4 ⫽k 4* ,

if P⫽0 then 兵 ␻ 共 k 1 兲 , ␻ 共 k 2 兲 , ␻ 共 k 3 兲 , ␻ 共 k 4 兲其 ⫽ 兵 ␻ 共 ⫺k 1 兲 , ␻ 共 ⫺k 2 兲 , ␻ 共 ⫺k 3 兲 , ␻ 共 ⫺k 4 兲其 .

165210-10

共58兲



A simplified definition of the spectral asymmetry is given by Eq. 共1兲, where k is presumed real. Regardless of the spectral symmetry or asymmetry, the evanescent modes 共those with k⫽k * ), if then exist, must comply with the relation

and P at ␻ ⫽ ␻ 0 . Those relations require ␨ 0 to be a triple root of the characteristic polynomial F( ␨ ) at ␻ ⫽ ␻ 0 , i.e.,

兵 ␻ 共 k 1 兲 , . . . 其 ⬅ 兵 ␻ 共 k *1 兲 , . . . 其

In view of Eqs. 共52兲 and 共44兲, the values ␨ 0 and ␨ 1 are related by

共59兲

following from Eq. 共44兲. A specific numerical example of asymmetric electromagnetic spectrum is shown in Fig. 1. The physical parameters of the corresponding periodic stack are chosen so that at a certain frequency ␻ 0 the dispersion relation ␻ (k) of one of the spectral branches develops a stationary inflection point. The corresponding frequency is associated with the electromagnetic unidirectionality. In the next subsection, we take a closer look at this particular situation. D. Stationary inflection point

The dispersion relation ␻ (k) of an arbitrary periodic stack is determined by the characteristic equation 共41兲, where the coefficients R, Q, and P are functions of the frequency ␻ . Using the characteristic equation 共41兲, we can define the stationary inflection point ␨ 0 ⫽exp(ik0L) in Eq. 共3兲 as one satisfying

⬙ 共 ␨ 0 兲 ⫽0 F 共 ␨ 0 兲 ⫽0, F ␨⬘ 共 ␨ 0 兲 ⫽0, F ␨␨

F 0 共 ␨ 兲 ⫽ ␨ 4 ⫹ 共 R 0 ⫺i P 0 兲 ␨ 3 ⫹Q 0 ␨ 2 ⫹ 共 R 0 ⫹i P 0 兲 ␨ ⫹1 ⫽ 共 ␨ ⫺ ␨ 1 兲共 ␨ ⫺ ␨ 0 兲 3 ⫽0.

共63兲

k 1 ⬅⫺3k 0 , Im k 0 ⫽Im k 1 ⫽0.

共64兲

A small deviation of the frequency ␻ from its special value ␻ 0 changes the coefficients R 0 , Q 0 , and P 0 in Eq. 共62兲 and removes the triple degeneracy of the solution ␨ 0 . Taking into account Eqs. 共60兲 and 共61兲, we have

␨ ⫺ ␨ 0 ⬇⫺ 共 6 兲

1/3

冉

⳵ F/ ⳵␻ ⳵ 3 F/ ⳵␨ 3

冊

1/3

共 ␻ ⫺ ␻ 0 兲 1/3␰ , ␨⫽␨0 ,␻⫽␻0

where ␰ ⫽1,e 2 ␲ i/3,e ⫺2 ␲ i/3,

共65兲

or, in terms of the quasimomentum k,

共60兲 k⫺k 0 ⬇

冉

1 ␻ ⵮共 k 0 兲 6

冊

⫺1/3

共 ␻ ⫺ ␻ 0 兲 1/3␰ ,

where ␰ ⫽1,e i(2 ␲ /3) ,e ⫺i(2 ␲ /3) .

共61兲

Equations 共60兲 impose certain relations upon the values R 0 , Q 0 , and P 0 of the frequency dependent coefficients R, Q ,

冦

␨ 1 ⫽ ␨ ⫺3 0 , 兩 ␨ 0 兩 ⫽ 兩 ␨ 1 兩 ⫽1 or, equivalently,

with an additional condition

⵮ 共 ␨ 0 兲 ⫽0. F ␨␨␨

共62兲

共66兲

We can also rearrange Eq. 共66兲 in a different form, which is actually used for further references,

k ex ⬇k 0 ⫹6 1/3关 ␻ ⵮ 共 k 0 兲兴 ⫺1/3共 ␻ ⫺ ␻ 0 兲 1/3,

冑3 1 k e v ⬇k 0 ⫹ 共 6 兲 1/3关 ␻ ⵮ 共 k 0 兲兴 ⫺1/3共 ␻ ⫺ ␻ 0 兲 1/3⫹i 6 1/3关 ␻ ⵮ 共 k 0 兲兴 ⫺1/3兩 ␻ ⫺ ␻ 0 兩 1/3, 2 2 冑3

共67兲

1 k EV ⬇k 0 ⫹ 共 6 兲 1/3关 ␻ ⵮ 共 k 0 兲兴 ⫺1/3共 ␻ ⫺ ␻ 0 兲 1/3⫺i 6 1/3关 ␻ ⵮ 共 k 0 兲兴 ⫺1/3兩 ␻ ⫺ ␻ 0 兩 1/3. 2 2

The real quasimomentum k ex in Eq. 共67兲 relates to the extended mode ⌿ ex (z), which turns into the frozen mode at ␻ ⫽ ␻ 0 . The other two solutions, k e v and k EV ⫽k * e v , correspond to a pair of evanescent modes, ⌿ e v (z) and ⌿ EV (z), with positive and negative imaginary parts, respectively. Those modes are truly evanescent 共i.e., have Im k⫽0) only if ␻ ⫽ ␻ 0 . But it does not mean that at ␻ ⫽ ␻ 0 , the eigenmodes ⌿ e v (z) and ⌿ EV (z) become extended! 165210-11

Eigenmodes at frequency ␻ 0 of stationary inflection point

Consider four eigenvectors, ⌽ k 1 ⫽⌿ k 1 共 0 兲 , ⌽ k 2 ⫽⌿ k 2 共 0 兲 , ⌽ k 3 ⫽⌿ k 3 共 0 兲 , ⌽ k 4 ⫽⌿ k 4 共 0 兲 ,

共68兲



of the transfer matrix T L from Eq. 共36兲 in the vicinity of stationary inflection point. As long as ␻ ⫽ ␻ 0 , four eigenvectors 共68兲 comprise two extended and two evanescent Bloch solutions. One of the extended modes 共say, ⌽ k 1 ) corresponds to the nondegenerate real root ␨ 1 ⫽e ik 1 L of the characteristic equation with the negative group velocity u(k 1 )⫽ ␻ ⬘ (k 1 ) ⬍0, as shown in Fig. 1. This solution, relating to the backward propagating mode, is of no interest for us. The other three eigenvectors of T L correspond to three nearly degenerate roots 共65兲. As ␻ approaches ␻ 0 , those three eigenvectors not only become degenerate, but they also become colinear, ⌽ k 2 → ␣ 2,4⌽ k 4 , ⌽ k 3 → ␣ 3,4⌽ k 4 as ␻ → ␻ 0 ,

共69兲

where ␣ 2,4 and ␣ 3,4 are complex scalars. The latter important feature relates to the fact that at ␻ ⫽ ␻ 0 , the matrix T L ( ␻ 0 ) has a nontrivial Jordan canonical form,

T L 共 ␻ 0 兲 ⫽U

冋

␨1

0

0

0

0

␨0

1

0

0

0

␨0

1

0

0

0

␨0

册

⳵ ⳵ ⌿ k 共 z 兲 ⫽e ikz ␺ k 共 z 兲 ⫹iz ␺ k 共 z 兲 e ikz ⳵k ⳵k and

⳵2

⳵2

⳵k

⳵k

U ⫺1 ,

共75兲

¨ k 共 z 兲 ⫹iz⌿ ˙ k 共 z 兲 ⫺z 2 ⌿ k 共 z 兲 , ⌿ 0,2共 z 兲 ⫽⌿ 0 0 0

˙ k 共 z 兲⫽ ⌿ 0

⫽

冉

冉

⳵ ␺ 共z兲 ⳵k k ⳵2 ⳵k2

冊

␺ k共 z 兲

¨ k 共z兲 e ik 0 z and ⌿ 0 k⫽k 0

冊

e ik 0 z k⫽k 0

are auxiliary Bloch functions 共not eigenmodes兲. Observe that only the first of the three solutions 共75兲 is a canonical Bloch eigenmode 关the frozen mode ⌿ k 0 (z)]. The other two solutions diverge as the first and the second power of z, respectively. They are referred to as general Floquet modes. Deviation of the frequency ␻ from ␻ 0 removes the triple degeneracy 共70兲 of the matrix T L , as can be seen from Eq. 共65兲. The modified matrix T L can now be reduced to a diagonal form with the set 共68兲 of four eigenvectors comprising two extended and two evanescent Bloch solutions. III. SEMI-INFINITE UNIDIRECTIONAL STACK

共72兲

⳵ ⳵ ⳵2 ˆ ⌿ k共 z 兲 ⌿ k共 z 兲 ⫽ ␻ 共 k 兲 ⌿ k共 z 兲 ; M ⳵k ⳵k ⳵k2

⳵k2

⳵ ␺ 共 z 兲 ⫺z 2 ␺ k 共 z 兲 e ikz ⳵k k 共74兲

where

Assume that the dispersion relation ␻ (k) in Eq. 共72兲 has a stationary inflection point 共3兲 at k⫽k 0 . Differentiating Eq. 共72兲 with respect to k gives, in view of Eq. 共3兲,

⫽␻共 k 兲

␺ k 共 z 兲 ⫹ize ikz

⌿ k 0共 z 兲 ,

of the reduced Maxwell equation 共21兲. By definition

⳵2

2

ˆ with the same eigenvalue ␻ 0 . are also eigenmodes of M Therefore all three solutions 共71兲, 共73兲, and 共74兲 are eigenˆ with the same eigenvalue ␻ 0 . For further refmodes of M erences we recast those three eigenmodes in the following form:

共70兲

⌿ k 共 z 兲 ⫽ ␺ k 共 z 兲 e ikz , where ␺ k 共 z⫹L 兲 ⫽ ␺ k 共 L 兲 , Im k⫽0 共71兲

ˆ at k⫽k 0 : M

⌿ k 共 z 兲 ⫽e ikz 2

˙ k 共 z 兲 ⫹iz⌿ k 共 z 兲 , ⌿ 0,1共 z 兲 ⫽⌿ 0 0

and therefore cannot be diagonalized 共see, for example, Ref. 17兲. It is shown rigorously in Appendix B, that the very fact that the T L eigenvalues display the singularity 共65兲 at ␻ ⫽ ␻ 0 implies that the matrix T L ( ␻ 0 ) has the canonical form 共70兲. One of the consequences of Eq. 共70兲 is that the matrix T L ( ␻ 0 ) has only two 共not four!兲 eigenvectors: 共1兲 ⌽ k 1 , corresponding to the nondegenerate root ␨ 1 , and 共2兲 ⌽ k 0 , corresponding to the triple root ␨ 0 and describing the frozen mode. The other two solutions of the Maxwell equation 共21兲 at ␻ ⫽ ␻ 0 are general Floquet eigenmodes which do not reduce to the canonical Bloch form. Yet, they can be related to the frozen mode ⌿ k 0 (z). Indeed, following the standard procedure 共see, for example, Ref. 18兲, consider an extended Bloch solution

ˆ ⌿ k共 z 兲 ⫽ ␻ 共 k 兲 ⌿ k共 z 兲 . M

共73兲

A. Transmittance and reflectance of a semi-infinite stack

Consider plane electromagnetic wave ⌿ I (z) impinging normally on the surface of a unidirectional semi-infinite slab, as shown in Fig. 5. In vacuum 共at z⬍0), the electromagnetic field ⌿ L (z) is a superposition of the incident and reflected waves at z⬍0: ⌿ L 共 z 兲 ⫽⌿ I 共 z 兲 ⫹⌿ R 共 z 兲

共76兲

where

⌿ k共 z 兲 .

⌿ I 共 z 兲 ⫽⌽ I exp

This implies that at k⫽k 0 , both functions 165210-12

冉冊

冉

冊

i␻ i␻ z , ⌿ R 共 z 兲 ⫽⌽ R exp ⫺ z , c c

共77兲



冋册 E I,x

⌽ I ⫽⌿ I 共 0 兲 ⫽

E I,y

⫺E I,y

冋册

B. Overview of the results

E R,x

, ⌽ R ⫽⌿ R 共 0 兲 ⫽

E R,y

E R,y

⫺E R,x

E Ix

, 共78兲

ជ R are complex vectors describing two elliptically Eជ I and E polarized waves. The transmitted wave ⌿ T (z) inside the stack is a superposition of two Bloch eigenmodes, at z⬎0: ⌿ T 共 z 兲 ⫽⌿ 1 共 z 兲 ⫹⌿ 2 共 z 兲 .

共79兲

The eigenmodes ⌿ 1 (z) and ⌿ 2 (z) can be both extended, one extended and one evanescent, or both evanescent, depending on which of the three cases 共46兲, 共48兲, or 共50兲 we are dealing with. In particular, if the frequency ␻ lies within the range ␻ a ⬍ ␻ ⬍ ␻ b in Fig. 1, we have the situation 共48兲, and the transmitted electromagnetic wave ⌿ T (z) is a superposition of the extended Bloch eigenmode ⌿ ex (z) with group velocity u⬎0 and the evanescent mode ⌿ e v (z) with Im k ⬎0, at z⬎0: ⌿ T 共 z 兲 ⫽⌿ ex 共 z 兲 ⫹⌿ e v 共 z 兲 .

共80兲

The only exception to Eq. 共80兲 is when the frequency ␻ exactly coincides with the frequency ␻ 0 of the frozen mode. In such a case, ⌿ T (z) is a linear combination of the Floquet eigenmodes 共75兲, one of which is extended 关the frozen mode ⌿ k 0 (z)] and the other two cannot be expressed in canonical Bloch form 共34兲. In what follows we assume that ␻ can be arbitrarily close but not equal to ␻ 0 , unless otherwise is explicitly stated. Extended and evanescent modes inside a periodic gyrotropic medium are defined by Eq. 共36兲. Knowing the Bloch eigenmodes inside the slab and using the standard electromagnetic boundary conditions ⌽ T ⫽⌽ I ⫹⌽ R

共81兲

at the slab surface at z⫽0, one can express the amplitudes ⌽ T and ⌽ R of transmitted and reflected waves in terms of the amplitude and polarization ⌽ I of the incident wave. This gives us the transmittance and reflectance coefficients of semi-infinite slab, as well as the electromagnetic field distribution ⌿ T (z) inside the slab, as functions of the incident wave polarization. The transmittance ␶ e and reflectance r e of semi-infinite slab are defined as

␶ e⫽

S共 ⌽T兲 , S共 ⌽I兲

␳ e ⫽⫺

S共 ⌽R兲 ; S共 ⌽I兲

␶ e ⫹ ␳ e ⫽1,

共82兲

where S共 ⌽ 兲⫽

A general idea of what happens when a plane electromagnetic wave of the frequency ␻ close to the frozen mode frequency ␻ 0 impinges on the surface of a semi-infinite unidirectional slab, is provided by the numerical examples shown in Figs. 7 and 8. First, the transmittance ␶ e of the semi-infinite unidirectional slab remains finite within the frequency range ␻ a ⬍ ␻ ⬍ ␻ b , including the frequency ␻ 0 of the frozen mode, as seen in in Fig. 7. By contrast, the transmittance ␶ e of any semi-infinite slab always vanishes in the vicinity of a band edge 共see, for example, the vicinity of ␻ ⫽ ␻ b in Fig. 7兲. That the incident wave with the frequency ␻ 0 can freely enter a semi-infinite unidirectional slab, in spite of the fact that the wave group velocity inside the slab vanishes at ␻ ⫽ ␻ 0 , has far-reaching implications. Second, the field amplitude inside unidirectional slab can rise by several orders of magnitude in the vicinity of the frozen mode frequency ␻ 0 , as shown in Figs. 8共a兲 and 10. This remarkable feature will be discussed in great detail later in this section. Third, in the vicinity of ␻ 0 , the density of mode has much stronger anomaly compared to that of the vicinity of a band edge frequency. This makes all the effects associated with the frozen mode much more robust. Finally, the transmittance as well as the reflectance coefficients develop a cusplike singularity at ␻ ⫽ ␻ 0 ; the magnitude and the sign of this singularity being dependent on the polarization of the incident wave. In particular, if the incident wave polarization is chosen so that only a single extended mode ⌿ ex (z) continues inside the slab 关no evanescent contribution to ⌿ T (z)], then the transmittance ␶ e at ␻ ⫽ ␻ 0 drops down to zero, as shown in Fig. 7共d兲. But, if the incident wave polarization is orthogonal to the previous one 关see Fig. 7共c兲兴, the transmittance of the unidirectional slab is maximal. The explanation for such an unusual behavior is given further in this section. Of course, if the incident wave polarization is chosen so that only a single evanescent mode ⌿ e v (z) continues inside the slab 关no extended contribution to ⌿ T (z)], the transmittance of any semi-infinite slab is strictly zero regardless of the frequency ␻ . C. Frequency dependence of electromagnetic field amplitude inside unidirectional slab

According to Eq. 共80兲, the transmitted wave ⌿ T (z) inside the stack is a superposition of one extended 共nearly frozen兲 mode ⌿ ex (z) and one evanescent mode ⌿ e v (z). Since evanescent modes do not transfer energy, the extended mode ⌿ ex (z) is solely responsible for the energy flux inside the stack. The energy density W ex associated with the extended ⌿ ex (z) can be expressed in terms of its group velocity u(k)⫽ ␻ ⬘ (k) and the energy density flux S(⌽ ex ), W ex ⫽S 共 ⌽ ex 兲 / ␻ ⬘ 共 k 兲 , where S 共 ⌽ ex 兲 ⫽S 共 ⌽ T 兲 ⫽ ␶ e S 共 ⌽ I 兲 . 共83兲

c 具 E H ⫺E y H x 典 4␲ x y

is the energy density flux averaged over the period of oscillations.

In line with Eq. 共3兲, in the vicinity of the frozen mode frequency

165210-13



1 ␻ 共 k 兲 ⫺ ␻ 0 ⬇ ␻ ⵮ 共 k 0 兲共 k⫺k 0 兲 3 6

Im k e v ⬇

冑3

6 1/3关 ␻ ⵮ 共 k 0 兲兴 ⫺1/3兩 ␻ ⫺ ␻ 0 兩 1/3.

2

共89兲

Plugging Eq. 共89兲 in Eq. 共88兲 gives

that gives 1 6 2/3 ␻ ⬘ 共 k 兲 ⬇ ␻ ⵮ 共 k 0 兲共 k⫺k 0 兲 2 ⬇ 关 ␻ ⵮ 共 k 0 兲兴 1/3共 ␻ ⫺ ␻ 0 兲 2/3. 2 2 共84兲

at z⬎0:

冋

兩 ⌿ e v 共 z 兲兩 ⬇ 兩 ⌽ e v 兩 1⫺z

Plugging Eq. 共84兲 into Eq. 共83兲 yields W ex ⬇

2 6

2/3

共85兲

where S I ⫽S(⌽ I ) is a fixed intensity of the incident wave, ␶ e is the transmittance coefficient 共82兲 depending on the incident wave polarization. Formula 共85兲 implies that the amplitude ⌽ ex of the extended nearly frozen mode inside the stack diverges in the vicinity of the stationary inflection point ⌽ ex ⬃ 冑W ex ⬃ 冑␶ e S I 关 ␻ ⵮ 共 k 0 兲兴

兩 ␻ ⫺ ␻ 0兩

⫺1/3

6 1/3关 ␻ ⵮ 共 k 0 兲兴 ⫺1/3

⌽ ex ⬇⫺⌽ e v ⬃ 共 ␻ ⫺ ␻ 0 兲 ⫺1/3 as ␻ → ␻ 0

共87兲

so that ⌽ T ⫽⌽ ex ⫹⌽ e v remains limited. The expression 共87兲 is in compliance with the earlier made statement 共69兲 that the column vectors ⌽ ex and ⌽ e v become colinear as ␻ → ␻ 0 . The numerical illustration of the behavior of the field amplitudes 兩 ⌽ ex 兩 2 , 兩 ⌽ e v 兩 2 and 兩 ⌽ T 兩 2 ⫽ 兩 ⌽ ex ⫹⌽ e v 兩 2 at the slab surface is illustrated in Figs. 9共a兲–共c兲, respectively. D. Space distribution of electromagnetic field inside unidirectional slab

Since ⌿ ex (z) is an extended Bloch eigenmode, its amplitude 兩 ⌿ ex (z) 兩 remains constant at z⬎0, while the amplitude of the evanescent contribution ⌿ e v (z) to the resulting field ⌿ T (z) decays as 兩 ⌿ e v 共 z 兲兩 ⫽ 兩 ⌽ e v 兩 e ⫺zIm k e v .

共88兲

Therefore, as the distance z from the unidirectional slab boundary increases, the destructive interference of the extended and evanescent modes becomes ineffective, and at z Ⰷ(Im k e v ) ⫺1 the only remaining contribution to ⌿ T (z) is the extended nearly frozen mode ⌿ ex (z) with huge and independent of z amplitude 共86兲. Let us consider the above behavior in more detail. According to Eq. 共67兲, in the vicinity of ␻ ⫽ ␻ 0 ,

冉

z 2 兩 ␻ ⫺ ␻ 0 兩 2/3 共 ␻ ⵮ 共 k 0 兲兲 2/3

冊册

,

which together with Eqs. 共86兲 and 共87兲 yields the following asymptotic expression for the evanescent mode amplitude as function of ␻ and z: 兩 ⌿ ev共 z 兲兩 ⬇

冑␶ e S I 关 ␻ ⵮ 共 k 0 兲兴 1/6

⫺z

as ␻ → ␻ 0 . 共86兲

The divergence of the extended mode amplitude ⌽ ex in the vicinity of the frozen mode frequency ␻ 0 imposes a similar kind of behavior on the evanescent mode amplitude ⌽ e v . Indeed, the boundary condition 共81兲 requires that the resulting field amplitude ⌽ T ⫽⌽ ex ⫹⌽ e v at the slab boundary at z⫽0 remains limited to match the sum ⌽ L ⫽⌽ I ⫹⌽ R of the incident and reflected waves outside the stack at z⫽0. The relation 共81兲 together with Eq. 共86兲 imply that there is a destructive interference of the extended ⌽ ex and evanescent ⌽ e v modes at the stack boundary

at z⬎0:

2

⫻ 兩 ␻ ⫺ ␻ 0 兩 1/3⫹O

共 ␶ e S I 兲关 ␻ ⵮ 共 k 0 兲兴 ⫺1/3共 ␻ ⫺ ␻ 0 兲 ⫺2/3,

⫺1/6

冑3

冑3 2

冋

兩 ␻ ⫺ ␻ 0 兩 ⫺1/3

6 1/3关 ␻ ⵮ 共 k 0 兲兴 ⫺1/3⫹O

冉

z 2 兩 ␻ ⫺ ␻ 0 兩 1/3 共 ␻ ⵮ 共 k 0 兲兲 2/3

冊册

.

共90兲 Finally, plugging ⌿ ex (z) from Eq. 共86兲 and ⌿ e v (z) from Eq. 共90兲 into ⌿ T (z)⫽⌿ ex (z)⫹⌿ e v (z) yields 兩 ⌿ T 共 z 兲兩 ⬇ 兩 ⌽ T 兩 ⫹z

冑␶ e S I 关 ␻ ⵮ 共 k 0 兲兴

冑3 1/2

2

6 1/3

as ␻ → ␻ 0 , 共91兲

where, according to Eq. 共87兲, 兩 ⌽ T 兩 Ⰶ 兩 ⌽ ex 兩 ⬇ 兩 ⌽ e v 兩 .

The asymptotic expression 共91兲 for 兩 ⌿ T (z) 兩 is consistent with the eigenmode ⌿ 1,0(z) from Eq. 共75兲, which represents one of the two Floquet-type solutions for the Maxwell equation 共21兲 at ␻ ⫽ ␻ 0 . A numerical example of electromagnetic field distribution 兩 ⌿ T (z) 兩 2 inside the semi-infinite unidirectional slab for the frequency ␻ close to ␻ 0 is shown in Fig. 8, while the limiting case 共91兲 of ␻ ⫽ ␻ 0 is shown in Fig. 10. The relation 共91兲 implies that the resulting field amplitude 兩 ⌿ T (z) 兩 2 increases as the second power of the distance z from the slab surface. It reaches its maximum value of 兩 ⌿ ex (z) 兩 2 ⬃( ␻ ⫺ ␻ 0 ) ⫺2/3 at zⰇl, where l⫽ 共 Im k e v 兲 ⫺1 ⫽

2

冑3

6 ⫺1/3关 ␻ ⵮ 共 k 0 兲兴 1/3兩 ␻ ⫺ ␻ 0 兩 ⫺1/3. 共92兲

There are two exceptions, however, merging into a single one at ␻ ⫽ ␻ 0 . The first exception occurs when the elliptic polarization of the incident wave is chosen so that it produces just a single extended eigenmode ⌿ ex (z) inside the slab 关no evanescent contribution to ⌿ T (z)]. In this case, ⌿ T (z) reduces to ⌿ ex (z), and its amplitude 兩 ⌿ T (z) 兩 remains limited and independent of z. As ␻ approaches ␻ 0 , the respective transmittance ␶ e vanishes in this case, as shown in Fig. 7共d兲. The second exception occurs when the elliptic polarization of the incident wave is chosen so that it produces

165210-14



just a single evanescent eigenmode ⌿ e v (z) inside the slab 关no extended contribution to ⌿ T (z)]. In such a case, ⌿ T (z) reduces to ⌿ e v (z), and its amplitude 兩 ⌿ T (z) 兩 decays exponentially with z in accordance with Eq. 共88兲. The respective transmittance coefficient ␶ e in this latter case is zero regardless of the frequency ␻ , because evanescent modes do not transfer energy. Importantly, as ␻ approaches ␻ 0 , the polarizations of the incident wave that produce either a sole extended or a sole evanescent mode become indistinguishable, which is a consequence of the property 共69兲 of the T L eigenvectors. If ⌽ I0 is such a polarization of the incident wave, the maximal transmittance is reached when the incident wave polarization is orthogonal to ⌽ I0 关see Fig. 7共c兲兴. Let us see what happens if the degree of spectral asymmetry of the unidirectional periodic stack is very small. In this situation the stationary inflection point k 0 , ␻ 0 in Fig. 1 is very close to the band edge k b , ␻ b . In such a case, the third derivative ␻ ⵮ (k 0 ) along with the transmittance ␶ e of the respective unidirectional slab at ␻ ⫽ ␻ 0 are also very small. At the same time, the ratio ␶ e / ␻ ⵮ (k 0 ), which according to Eq. 共91兲 determines the electromagnetic field distribution inside the slab at ␻ → ␻ 0 , remains finite even if the quantities ␶ e and ␻ ⵮ (k 0 ) vanish. This implies that at ␻ ⫽ ␻ 0 , the character of the field distribution shown in Fig. 10 does not change qualitatively even if

tionality does not manifest itself in the case of the backward incidence. Indeed, the evanescent contribution ⌿ EV (z) to ⌿ T (z) still displays a singularity at ␻ ⫽ ␻ 0 , although its amplitude now remains limited even at ␻ ⫽ ␻ 0 . Let us take a closer look at this situation. The complex wave vector k EV related to ⌿ EV (z) has negative imaginary part and is defined in Eq. 共67兲, k EV ⬇k 0 ⫹

Its singularity at ␻ ⫽ ␻ 0 leads to a cusplike anomaly in frequency dependence of the backward transmittance of semiinfinite unidirectional slab, similar to what we already saw in the case of forward incidence 共see Fig. 7兲. But there is a crucial difference: the propagating mode amplitude 兩 ⌿ EX (z) 兩 ⫽ 兩 ⌽ EX 兩 now remains limited in the whole frequency range ␻ a ⬍ ␻ ⬍ ␻ b , including the frozen mode frequency ␻ 0 , as shown in Fig. 11. By contrast, in the case of forward incidence, the propagating mode amplitude 兩 ⌿ ex (z) 兩 ⫽ 兩 ⌽ ex 兩 along with the field amplitude 兩 ⌿ T (z) 兩 inside the stack rises enormously in the vicinity of the frozen mode frequency ␻ 0 , as shown in Fig. 9共a兲. This striking difference between the cases of forward and backward incidence can be attributed to the frozen mode. IV. A FINITE UNIDIRECTIONAL SLAB

␻ ⬘ 共 k 0 兲 ⫽ ␻ ⬙ 共 k 0 兲 ⫽ ␻ ⵮ 共 k 0 兲 ⫽0, ␻ ⵮⬘ 共 k 0 兲 ⫽0. Such a situation, however, corresponds to a degenerate band edge, rather than to a stationary inflection point 共3兲. E. Backward wave incidence on a semi-infinite unidirectional slab

Consider now electromagnetic wave incident on the surface of the same unidirectional slab from the opposite direction, as shown in Fig. 6. Such a situation is similar to that of the forward incidence on the reversed slab, which can be obtained from the original unidirectional slab in Fig. 2 by changing the sign of the F layers magnetization or by changing the sign of the misalignment angle ␸ ⫽ ␸ 1 ⫺ ␸ 2 of the anisotropic dielectric layers. Except for some obvious modifications involving the substitution z→⫺z, formulas 共76兲–共83兲 still apply here. In particular, if the frequency ␻ lies within the range ␻ a ⬍ ␻ ⬍ ␻ b in Fig. 1, the transmitted electromagnetic wave ⌿ T (z) inside the slab is a superposition of the extended Bloch eigenmode ⌿ EX (z) with the group velocity u⬍0 and the evanescent mode ⌿ EV (z) with Im k⬍0, at z⬍0: ⌿ T 共 z 兲 ⫽⌿ EX 共 z 兲 ⫹⌿ EV 共 z 兲 .

6 1/3 关 ␻ ⵮ 共 k 0 兲兴 ⫺1/3关共 ␻ ⫺ ␻ 0 兲 1/3⫺i 冑3 兩 ␻ ⫺ ␻ 0 兩 1/3兴 . 2

共93兲

This expression is similar to Eq. 共80兲, except that it involves the other pair of the four Bloch eigenmodes. In the case of backward incidence, the nearly frozen mode ⌿ ex (z) does not contribute to ⌿ T (z) inside semi-infinite slab. Instead, the extended contribution to the resulting transmitted electromagnetic field ⌿ T (z) is now ⌿ EX (z), which remains a regular extended mode with finite negative group velocity even at ␻ ⫽ ␻ 0 . It does not mean, however, that the slab unidirec-

Strictly speaking, the concept of unidirectionality applies to infinite or semi-infinite periodic stacks. But in reality, if we have a finite slab, which is a sufficiently large fragment of a periodic unidirectional stack, the results of the previous section can still be relevant. Let N be the number of the primitive cells in the slab, so that the slab thickness D is equal to LN. The approximation of infinite or semi-infinite stack applies if 1Ⰶ 共 L⌬k 兲 ⫺1 ⰆN,

共94兲

where ⌬k is the spectral width of the wave packet. In such a case, the interference of the pulses produced by internal reflections from the two opposite slab boundaries can be ignored. The results of the previous section relate to this particular case. In this section we consider a different situation when 1ⰆNⰆ 共 L⌬k 兲 ⫺1 .

共95兲

In particular, we can refer to the limiting case ⌬k⫽0 of a strictly monochromatic incident wave. In this latter case the approximation of infinite or semi-infinite slab does not hold for any finite N, due to multiple internal reflections form the two slab boundaries. The electromagnetic field ⌿ T (z) inside the slab is now a superposition of all four Bloch eigenmodes ⌿ k i (z), i⫽1,2,3,4 for any given frequency ␻ regardless of the direction of the incident wave propagation outside the slab. By contrast, in the case 共94兲 of semi-infinite slab, there are only two Bloch contributions 共79兲 to ⌿ T (z). At the same time, we expect some noticeable electromagnetic abnormalities even in the limiting case 共95兲, provided that the number N of the elementary fragments L in the slab is large enough.

165210-15



FIG. 13. Forward 共a兲 and backward 共b兲 tramittance of the finite unidirectional slab with N ⫽32. The frequency ␻ lies in the vicinity of the lowest band edge ␻ b in Fig. 1.

Transmittance of a finite unidirectional slab

The transmittance of an arbitrary finite slab can be expressed directly in terms of the transfer matrix T N of the slab, which in our case is defined by T N⫽共 T L 兲N.

共96兲

⌿ 共 D 兲 ⫽T N ⌿ 共 0 兲

共97兲

Indeed, the relation

together with the pair of boundary conditions ⌿ 共 0 兲 ⫽⌿ I 共 0 兲 ⫹⌿ R 共 0 兲 ,

⌿ 共 D 兲 ⫽⌿ P 共 D 兲

共98兲

allow us to express both the reflected wave ⌿ R (0) and the wave ⌿ P (D) passed through the slab, in terms of a given incident wave ⌿ I (0) from Eq. 共78兲 and the elements of the transfer matrix T N . It also gives the transmittance/reflectance coefficients of the slab defined as

␶ N⫽

兩 ⌿ P共 D 兲兩 2 兩 ⌿ I共 0 兲兩 2

,

␳ N⫽

兩 ⌿ R共 0 兲兩 2 兩 ⌿ I共 0 兲兩 2

;

␶ N ⫹ ␳ N ⫽1, 共99兲

respectively. The above procedure is commonly used for computation of the transmittance/reflectance coefficients of magnetic layered structures 共see, for example, Refs. 13–16 and references therein兲. Notice that as long as we are dealing with strictly monochromatic incident wave (⌬k⫽0), the transmittance/reflectance coefficients 共82兲 of a semi-infinite slab cannot be viewed as the limiting case N→⬁ of the transmittance/reflectance coefficients 共99兲 of a finite slab. Since the transmittance computation for a finite slab with a given transfer matrix T N is a well-established procedure,

we skip the details and turn to the physical results. If a slab is composed of just a few elementary cells L in Fig. 2, its transmittance does not show any peculiarities in the vicinity of the frozen mode frequency ␻ 0 . As the number N increases, the electromagnetic abnormalities in the vicinity of the frozen mode frequency become more and more distinct. In Figs. 12 and 13 we present some numerical results for the transmittance of a finite unidirectional slab comprising N ⫽32 identical elementary fragments L. This number of layers appears to be large enough to display all the qualitative features characteristic of a very thick unidirectional slab. The most distinguishable new feature is that the forward 共left-toright兲 and the backward 共right-to-left兲 transmittance coefficients do show a strong abnormality in the vicinity of the frozen mode frequency ␻ 0 . In particular, if the elliptic polarization of the incident wave coincides with that of the maximal transmittance of the respective semi-infinite slab 关see Fig. 7共c兲兴, the finite slab becomes totally transparent, as shown in Fig. 12共a兲. A small difference between the frequency of total transmittance and ␻ 0 is due to a finite thickness of the slab. At frequencies not too close to ␻ 0 , the electromagnetic properties of a unidirectional slab are not much different from those of regular magnetic stacks 共see, for example,13–16 and references therein兲. In particular, at certain polarizations of the incident wave, a finite slab displays both, forward and backward resonant transmittance even in the close proximity of the band edges, as shown in Fig. 13. V. SUMMARY

As we have shown, the phenomenon of unidirectionality in magnetic photonic crystals is always associated with the

165210-16



existence of special propagating mode with zero group velocity ␻ k⬘ (k) and its derivative ␻ ⬙kk (k). We call it the frozen mode. At first glance, the fact that the frozen mode has zero group velocity will bring some similarity between the vicinity of the frozen mode frequency 共at ␻ ⬇ ␻ 0 ), and the vicinity of the photonic band edge 共say, at ␻ ⬇ ␻ b ). Indeed, in either situation the propagating electromagnetic wave inside the periodic medium slows down dramatically, although in the frozen mode case the slowdown occurs only in one of the two opposite directions 共the unidirectionality兲. But in fact, the dissimilarity between the two situations does not reduce just to the phenomenon of unidirectionality. The most graphic manifestation of the fundamental difference between the vicinity of the frozen mode frequency ␻ 0 and the vicinity of a band-gap frequency ␻ b is provided by the simple and important case of electromagnetic wave incidence on the surface of a semi-infinite slab shown in Fig. 5. In a broad vicinity of the frozen mode frequency, including the point ␻ ⫽ ␻ 0 , the incident radiation enters the semiinfinite slab with little reflectance. By contrast, at ␻ ⬇ ␻ b the same semi-infinite slab reflects 100% of the incident radiation. This crucial difference is illustrated in Figs. 7共a兲–共c兲. In fact, the only way to transmit the radiation at frequency ␻ ⬇ ␻ b inside the slab is to make the slab thin enough to ensure strong interference after multiple reflections from the two slab boundaries 共see, for example, Ref. 21 and references therein兲. What happens in a photonic crystal at frequencies close to the frozen mode frequency ␻ 0 is that the pulse freely enters the slab, where it slows down by, say, two or three orders of magnitude and increases in amplitude proportionally. Then the pulse slowly continues through the slab without losing its distinct individuality until it reaches the opposite boundary or gets converted or absorbed inside the slab. The fact that in the vicinity of the stationary inflection point 共3兲共i.e., at ␻ ⬙ (k) vanishes, further contrib⬇ ␻ 0 ) the space dispersion ␻ kk uting to the pulse stability. Nothing like that can occur in any regular photonic crystal, not supporting the frozen mode. In addition, the electromagnetic density of mode displays a much stronger anomaly at ␻ ⬇ ␻ 0 compared to any other location in the Brillouin zone including the band edges. The latter circumstance must facilitate the observation and utilization of the frozen mode phenomena. The above unique features, in a combination with the relative simplicity of the multilayered structures, can make unidirectional photonic crystals very attractive for practical purposes. This may include: • various nonlinear applications 共see, for example, Refs. 22 and 21, and references therein兲, which can take advantage of huge amplitude of the frozen mode, in a combination with high transmittance and high density of modes at the respective frequency; • tunable delay lines, utilizing low group velocity of the ⬙ ⬇0) and high frozen mode, as well as its low dispersion ( ␻ kk transmittance of the slab; • electromagnetic nonreciprocal devices, utilizing the phenomenon of unidirectionality itself.

ACKNOWLEDGMENT AND DISCLAIMER

The efforts of A. Figotin and I. Vitebskiy were sponsored by the Air Force Office of Scientific Research, Air Force Materials Command, USAF, under Grant No. F49620-01-10567. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research or the U.S. Government. APPENDIX A: GYROTROPIC STACK WITH THREE-LAYERED CELL

Having studied numerically a number of periodic magnetic stacks with bulk spectral asymmetry, we come to the following conclusion. As long as we restrict ourselves to the lowest spectral band, the electromagnetic dispersion relations with stationary inflection point computed for different stacks appear to be qualitatively similar to each other and to what is shown in Fig. 1. In addition to this, since our prime interest here is with the vicinity of the frozen mode frequency ␻ 0 , all essential electromagnetic features prove to be quite universal and dependent on a single dimensionless parameter ␾ from Eq. 共4兲. In the case of strong spectral asymmetry, ␾ is of the order of magnitude of unity. This circumstance allows us to use any particular numerical example to obtain a complete picture of what is going on in unidirectional photonic crystals in the vicinity of the frozen mode frequency. Example considered in this section represents the simplest and, perhaps, the most practical design of a periodic layered structure with the property of bulk spectral asymmetry 共1兲. This array, shown in Fig. 2, is similar to that considered in Ref. 10. As already noted, a particular choice of the physical parameters of the stack does not matter, as long as it provides a certain value of ␾ . The A layers are described by the following reduced property tensors: ␧ˆ A⫽

␮ˆ A⫽

冋

冋

册冋册冋

␧ xx

␧ xy

␧ xy

␧yy

␮ xx

␮ xy

␮ xy

␮yy

⫽

⫽

␧⫹ ␦ cos 2 ␸

␦ sin 2 ␸

␦ sin 2 ␸

␧⫺ ␦ cos 2 ␸

册

,

␮ ⫹⌬ cos 2 ␸

⌬ sin 2 ␸

⌬ sin 2 ␸

␮ ⫺⌬ cos 2 ␸

册

. 共A1兲

ˆ A are presumed real. Parameters All components of ␧ˆ A and ␮ ␦ and ⌬ describe the anisotropy in the xy plane, while the angle ␸ defines the orientation of the common principle axes ˆ A in the xy plane. The misalignment angle ␸ 1 of ␧ˆ A and ␮ ⫺ ␸ 2 between the neighboring layers in Fig. 2 must be different from 0 and ␲ /2. All A layers are made of the same dielectric material and have the same thickness A . The four solutions for the Maxwell equation 共21兲 with material relations 共A1兲 are

165210-17



e iq 1 z ⌽ A1 ,e ⫺iq 1 z ⌽ A1 ,e iq 2 z ⌽ A2 ,e ⫺iq 2 z ⌽ A2 , where

⌽ A1 ⫽

冋册冋册 cos ␸

⫺sin ␸

sin ␸

cos ␸

⌽ A2 ⫽

,

⫺ ␩ 1 sin ␸

⫺ ␩ 2 cos ␸

␩ 1 cos ␸

␩ 1 ⫽ 冑共 ␧⫹ ␦ 兲共 ␮ ⫺⌬ 兲 ⫺1 ,

共A2兲

共A3兲

,

⫺ ␩ 2 sin ␸

ˆ 共 ␸ ,A 兲 W ˆ ⫺1 共 ␸ ,0兲 , Tˆ A 共 ␸ ,A 兲 ⫽W

␻ ␻ q 2 ⫽ n 2 ⫽ 冑共 ␧⫺ ␦ 兲共 ␮ ⫹⌬ 兲 , c c

冋

a⫽

共A4兲

共 cos ␸ 兲 e ⫺in 1 a

⫺ 共 sin ␸ 兲 e in 2 a

⫺ 共 sin ␸ 兲 e ⫺in 2 a

共 sin ␸ 兲 e in 1 a

共 sin ␸ 兲 e ⫺in 1 a

共 cos ␸ 兲 e in 2 a

共 cos ␸ 兲 e ⫺in 2 a

⫺ ␩ 1 共 sin ␸ 兲 e in 1 a

␩ 1 共 sin ␸ 兲 e ⫺in 1 a

⫺ ␩ 2 共 cos ␸ 兲 e in 2 a

␩ 2 共 cos ␸ 兲 e ⫺in 2 a

␩ 1 共 cos ␸ 兲 e in 1 a

⫺ ␩ 1 共 cos ␸ 兲 e ⫺in 1 a

⫺ ␩ 2 共 sin ␸ 兲 e in 2 a

␩ 2 共 sin ␸ 兲 e ⫺in 2 a

␻ A. c

共A8兲

冋

⑀

i␣

⫺i ␣

⑀

册

ˆ F⫽ ; ␮

冋

␮

i␤

⫺i ␤

␮

册

⌽ F1 ⫽

冋册冋册 ⫺i

⫺i

1

i␩1

, ⌽ F2 ⫽

␩1

⫺␩2

ˆ 共 F 兲W ˆ ⫺1 共 0 兲 , Tˆ F ⫽W

ˆ 共F兲 W

␻ 冑共 ⑀ ⫺ ␣ 兲共 ␮ ⫺ ␤ 兲 , c

⫽

共A10兲

共A14兲

冋

e in 1 f

e ⫺in 1 f

⫺ie in 2 f

⫺ie ⫺in 2 f

⫺ie in 1 f

⫺ie ⫺in 1 f

e in 2 f

e ⫺in 2 f

i ␩ 1 e in 1 f

⫺i ␩ 1 e ⫺in 1 f

⫺ ␩ 2 e in 2 f

␩ 2 e ⫺in 2 f

␩ 1 e in 1 f

⫺ ␩ 1 e ⫺in 1 f

⫺i ␩ 2 e in 2 f

i ␩ 2 e ⫺in 2 f

,

共A11兲

⫺i ␩ 2

共A12兲

␩ 1 ⫽ 冑共 ⑀ ⫹ ␣ 兲共 ␮ ⫹ ␤ 兲 ⫺1 , ␩ 2 ⫽ 冑共 ⑀ ⫺ ␣ 兲共 ␮ ⫺ ␤ 兲 ⫺1 .

共A13兲

册

,

共A15兲 f⫽

␻ ␻ ␻ q 1 ⫽ n 1 ⫽ 冑共 ⑀ ⫹ ␣ 兲共 ␮ ⫹ ␤ 兲 , q 2 ⫽ n 2 c c c ⫽

共A7兲

,

where 共A9兲

.

e iq 1 z ⌽ F1 ,e ⫺iq 1 z ⌽ F1 ,e iq 2 z ⌽ F2 ,e ⫺iq 2 z ⌽ F2 ,

1

册

ˆ (z) from Eq. 共25兲 Substituting the eigenmodes 共A10兲 into W and using the definition 共27兲 of the T matrix, we have the following expression for the transfer matrix Tˆ F of an individual F layer as a function of the layer thickness F:

The real parameters ␣ and ␤ in Eq. 共A9兲 are responsible for Faraday rotation. All F layers have the same thickness F. The four solutions for the Maxwell equation 共21兲 with material relations 共A9兲 are

where

共A6兲

where

共 cos ␸ 兲 e in 1 a

The F layers are ferromagnetic 共or ferrimagnetic兲 with ជ 0 parallel to the z direction; there is no inmagnetization M plane anisotropy in this case, ␧ˆ F⫽

共A5兲

ˆ (z) from Eq. Substituting the eigenmodes 共A2兲 into W 共25兲 and using the definition 共27兲 of the T matrix, we have the following expression for the transfer matrix Tˆ A of an individual A layer as a function of the layer thickness A and the misalignment angle ␸ :

␻ ␻ q 1 ⫽ n 1 ⫽ 冑共 ␧⫹ ␦ 兲共 ␮ ⫺⌬ 兲 , c c

ˆ 共 ␸ ,A 兲 ⫽ W

␩ 2 ⫽ 冑共 ␧⫺ ␦ 兲共 ␮ ⫹⌬ 兲 ⫺1 .

␻ F. c

共A16兲

ជ 0 , are The T atrices of F layers with two opposite signs of M related by transposition of the indices 1 and 2. Having the T matrices of both constitutive layers, one can obtain the explicit expression for the transfer matrix T L of the three-layered primitive cell in Fig. 2, T L 共 ␸ ,A,F 兲 ⫽T A 共 ␸ 1 ,A 兲 T A 共 ␸ 2 ,A 兲 T F 共 F 兲 .

共A17兲

Symbolic analysis of the transfer matrix T L ( ␸ ,A,F), as well as the corresponding characteristic equation 共38兲, has been carried out using the computer algebra package of ‘‘maple 7.’’

165210-18



We have also conducted a number of numerical experiments with this particular gyrotropic stack. When it comes to the vicinity of the frozen mode frequency, the general picture is universal, provided that the dimensionless parameter ␾ from Eq. 共4兲 is not too small. For this reason, all numerical illustrations in this paper refer to a single numerical set of material parameters of the stack chosen as follows: for the A layer:

n 1 ⫽5.1, ␩ 1 ⫽5.1, n 2 ⫽1.1, ␩ 2 ⫽1.1,

for the F layer:

共 ␨ ⫺ ␨ 0 兲 3 共 ␨ ⫺ ␨ 1 兲 ⫽0,

where

␨ 1 ⫽ ␨ ⫺3 0 , 兩 ␨ 1 兩 ⫽ 兩 ␨ 0 兩 ⫽1, ␨ 1 ⫽ ␨ 0 ,

n 1 ⫽22.023, ␩ 1 ⫽0.227 04, n 2 ⫽10.724, ␩ 2 ⫽0.466 25,

where the complex valued functions P J ( ␯ ), j⫽0,1,2,3 are analytic in ␯ in a vicinity of ␯ ⫽0. According to Eq. 共38兲, the frozen mode regime at ␯ ⫽0 can be ultimately characterized by the fact that for ␯ ⫽0 Eq. 共B3兲 takes the following special form

共A18兲

共B4兲

where ␨ 0 ⫽e ik 0 L corresponds to the frozen mode. If the characteristic equation 共B2兲 takes the special form 共B4兲 near ␯ ⫽0, then T L ( ␻ ) can be represented as follows:

with the misalignment angle ˆ 共␯兲 Tˆ L 共 ␻ 0 ⫹ ␯ 兲 ⫽U

␸ 1 ⫺ ␸ 2 ⫽ ␲ /4.

冋

␨ 1共 ␯ 兲

0

0

ˆ 共␯兲 Q

册

ˆ ⫺1 共 ␯ 兲 , U

共B5兲

The numerical values 共A18兲 are practically available at frequencies below 10 12Hz, but otherwise they are chosen randomly. On the other hand, having set the material parameters 共A18兲, we must find the exact values of the layer thicknesses so that at some frequency ␻ 0 the stack develops a stationary inflection point 共3兲 and therefore displays the property of unidirectionality. For the numerical values 共A18兲 we found

ˆ ( ␯ ) is a 3 is an analytic in ␯ complex valued function; Q ⫻3 matrix depending analytically on ␯ . In addition to that,

␳ 0 ⫽F/A⫽0.009 536 025 9,

ˆ 0 ⫹Q ˆ 1 ␯ ⫹•••, Q ˆ 0 ⫽ ␨ 0 Iˆ 3 ⫹D ˆ, ˆ 共 ␯ 兲 ⫽Q Q

⍀ 0 ⫽L ␻ 0 /c⫽0.607 676 756,

共A19兲

ˆ ( ␯ ) is an invertable 4⫻4 matrix depending analytiwhere U cally on ␯ ;

␨ 1 共 ␯ 兲 ⫽ ␨ 1 ⫹ ␰ 1 ␯ ⫹ ␰ 2 ␯ 2 ⫹•••

共B6兲

共B7兲

where Iˆ 3 is 3⫻3 identity matrix, and

K 0 ⫽k 0 L⫽2.632 925 94

ˆ ˆ 0 ⫽ ␨ 0 Î 3 ⫹D Q

where ␳ 0 is the required ratio of the layer thicknesses; ⍀ 0 is the dimensionless frozen mode frequency; and K 0 is the the dimensionless wave vector associated with the frozen mode. In all numerical graphs presented in this paper we use the dimensionless notations ␻ L/c and kL for the frequency and the wave vector, respectively.

ˆ0 is the spectral decomposition 共related to Jordan forms兲 of Q 26 ˆ being nilpotent matrix 共see Ref. 24, Sec. 6兲, i.e., with D

APPENDIX B: ANALYTIC PROPERTIES OF THE TRANSFER MATRIX Tˆ L IN A VICINITY OF THE FROZEN MODE FREQUENCY

ˆ 3 ⫽0. D

共B8兲

共B9兲

We would like to show that Eq. 共B8兲 is nontrivial in the ˆ ⫽0 and, in addition to that, sense that D ˆ 2 ⫽0. D

共B10兲

ˆ ( ␯ ) is Notice that the characteristic equation for Q

Consider the frequency-dependent 4⫻4 transfer matrix T L ( ␻ ) in a vicinity of the frozen mode frequency ␻ 0 ,

ˆ 共 ␯ 兲 ⫺ ␨ Î 3 兲 ⫽0, ␨ ⫽e ikL , det共 Q

Tˆ L 共 ␻ 兲 ⫽Tˆ L0 ⫹ ␯ Tˆ L1 ⫹•••, ␯ ⫽ ␻ ⫺ ␻ 0 ;

and, in view of Eqs. 共B2兲 and 共B4兲 it takes the following form:

Tˆ L0 ⫽Tˆ L 共 ␻ 0 兲 , Tˆ L1 ⫽Tˆ L⬘ 共 ␻ 0 兲 , . . . .

共B1兲

We assume the dependence of Tˆ L ( ␻ ) on ␻ to be analytic in the vicinity of ␻ ⫽ ␻ 0 . The following considerations are based on general facts from the analytic perturbation theory for the spectra of matrices.23 The characteristic equation 共38兲 for Tˆ L ( ␻ ) has the form det关 Tˆ L 共 ␻ 兲 ⫺ ␨ Î 4 兴 ⫽0,

␨ ⫽e

ikL

,

共B2兲

where Î 4 is the 4⫻4 identity matrix. Since T L ( ␻ ) is a 4 ⫻4 matrix, Eq. 共B2兲 can be recast as

␨ 4 ⫹ P 3 共 ␯ 兲 ␨ 3 ⫹ P 2 共 ␯ 兲 ␨ 2 ⫹ P 1 共 ␯ 兲 ␨ ⫹1⫽0,

共B3兲

共B11兲

共 ␨ ⫺ ␨ 0 兲 3 ⫹ 关 p 2 ␯ ⫹O 共 ␯ 2 兲兴共 ␨ ⫺ ␨ 0 兲 2

⫹ 关 p 1 ␯ ⫹O 共 ␯ 2 兲兴共 ␨ ⫺ ␨ 0 兲 ⫹ p 0 ␯ ⫽0,

共B12兲

where p 0 ⫽0

共B13兲

关according to Eq. 共66兲, p 0 ⫽6iL 3 / ␻ ⵮ (k 0 )]. In view of the Caley-Hamilton theorem 共see, for instance, Ref. 24, Sec. ˆ ( ␯ ) of the form 共B7兲 satisfies the characteristic equa6.2兲, Q tion 共B12兲. In other words, Eq. 共B12兲 holds if we substitute ˆ ( ␯ ) treating all other complex numbers as scalar matri␨ ⫽Q ces, i.e.,

165210-19



ˆ 共 ␯ 兲 ⫺ ␨ 0 Î 3 兴 3 ⫹p 2 ␯ 关 Q ˆ 共 ␯ 兲 ⫺ ␨ 0 Î 3 兴 2 ⫹p 1 ␯ 关 Q ˆ 共 ␯ 兲 ⫺ ␨ 0 Î 3 兴关Q ⫹p 0 Iˆ 3 ␯ ⫹O 共 ␯ 2 兲 ⫽0.

共B14兲

ˆ ( ␯ )⫽Q ˆ 0 ⫹Q ˆ 1 ␯ ⫹O( ␯ 2 ) in Eq. 共B14兲 and Now substituting Q taking in account Eq. 共B8兲 we single out the linear with respect to ␯ terms getting the following matrix equations: ˆ 1 ⫹D ˆQ ˆ 1D ˆ ⫹Q ˆ 1D ˆ ⫹p 2 D ˆ ⫹p 1 D ˆ ⫽⫺p 0 Iˆ 3 . ˆ Q D 2

2

ˆ ⫽0, which together with Eq. In view of Eq. 共B9兲, det D 共B17兲 implies that p 0 ⫽0, contradicting Eq. 共B13兲. Therefore ˆ has nontrivial Jordan ˆ 0 ⫽ ␨ 0 Î 3 ⫹D Eq. 共B10兲 is correct and Q ˆ 0 we have structure. In fact, in view of Eq. 共B9兲 Q

冋

2

共B15兲

Suppose now for the sake of argument that Eq. 共B10兲 does not hold, and hence D 2 ⫽0. Then Eq. 共B15兲 turns into ˆ ⫹p 1 D ˆ ⫽p 0 Î 3 , ˆQ ˆ 1D D

1

0

ˆ 0 ⫽Sˆ 0 0 Q 0

␨0

1 Sˆ ⫺1 0 ␨0

0

for some invertable Sˆ 0 . Notice also that

冋册

共B16兲

implying ˆQ ˆ 1D ˆ ⫹ p 1D ˆ 兲 ⫽det Ddet共 Q ˆ 1D ˆ ⫹p 1 兲 ⫽p 30 . det共 D

册

␨0

0

1

0

ˆ 共 ␯ 兲⫽ 0 Q ␯

0

1

0

0

共B18兲

is an exact solution to the matrix equation 共B17兲

E. Yablonovich, Phys. Rev. Lett. 58, 2059 共1987兲. J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals 共Princeton University Press, Princeton, 1995兲. 3 Photonic Band Gap Materials, edited by C. M. Soukoulis 共Kluwer Academic, Dordrecht, 1996兲. 4 Ph. Russell, T. Birks, and F. D. Lloyd-Lucas, Photonic Bloch Waves and Photonic Band Gaps 共Plenum, New York, 1995兲. 5 J. Rarity and C. Weisbuch, Microcavities and Photonic Band Gaps: Physics and Applications, Vol. 324 of NATO Advanced Study Institute Series E: Applied Sciences 共Kluwer Academic, Dordrecht, 1996兲. 6 Special Section on Electromagnetic Crystal Structures, Design, Synthesis, and Applications, 共IEEE/OSA兲, 1999, Vol. 17, pp. 1925–2424. 7 L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media 共Pergamon, New York, 1984兲. 8 A. Yariv and P. Yeh, Optical Waves in Crystals 共WileyInterscience, New York, 1984兲. 9 A. G. Gurevich and G. A. Melkov Magnetization Oscillations and Waves 共CRC Press, New York, 1996兲. 10 A. Figotin and I. Vitebsky, Phys. Rev. E 63, 066609 共2001兲. 11 H. How and C. Vittoria, Phys. Rev. B 39, 6823 共1989兲. 12 H. How and C. Vittoria, Phys. Rev. B 39, 6831 共1989兲. 13 M. Inoue and T. Fujii, J. Appl. Phys. 81, 5659 共1997兲. 14 M. Inoue and K. Arai, J. Appl. Phys. 83, 6768 共1998兲.

ˆ 3共 ␯ 兲 ⫽ ␯ I 3 . Q

共B19兲

I. Abdulhalim, J. Opt. A, Pure Appl. Opt. 2, 557 共2000兲. I. Abdulhalim, J. Opt. A, Pure Appl. Opt. 1, 646 共1999兲. 17 R. Bellman, Introduction to Matrix Analysis 共SIAM, Philadelphia, 1997兲. 18 E. Coddington and R. Carlson, Linear Ordinary Differential Equations 共SIAM, Philadelphia, 1997兲. 19 B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics 共John Wiley & Sons, Inc., New York, 1991兲. 20 A. Figotin, Yu. A. Godin, and I. Vitebsky, Phys. Rev. B 57, 2841 共1998兲. 21 G. D’Aguano, M. Centini, M. Scalora, et al., Phys. Rev. E 64, 016609 共2001兲. 22 M. Scalora, M. J. Bloemer, et al., Phys. Rev. A 56, 3166 共1997兲. 23 T. Kato, Perturbation Theory of Linear Operators 共Springer, New York, 1995兲. 24 P. Lancaster and M. Tismenetsky, The Theory of Matrices 共Academic, New York, 1985兲. 25 In the case of surface waves, the conditions for spectral asymmetry 共1兲 are generally less stringent compared to those of the bulk waves 共see, for example, Refs. 11 and 12 and references therein兲. Surface waves are beyond the scope of our investigation. 26 ˆ (z 2 ,z 1 ) from Eq. 共26兲 is called In some publications, the matrix T the propagation matrix or the state matrix.

1

15

2

16

165210-20