Magnetically induced topological transitions

Magnetically induced topological transitions of hyperbolic dispersion inbiaxial gyrotropic media

Vladimir R. Tuz1, a) and Volodymyr I. Fesenko1, 2, b)1)State Key Laboratory of Integrated Optoelectronics, College of Electronic Science and Engineering,International Center of Future Science, Jilin University, 2699 Qianjin Street, Changchun 130012,China2)Institute of Radio Astronomy of National Academy of Sciences of Ukraine, 4 Mystetstv Street, Kharkiv 61002,Ukraine

(Dated: 13 May 2020)

Magnetically induced topological transitions of isofrequency surfaces of bulk waves propagat-ing through an unbounded biaxial gyrotropic medium are studied. The medium is constructedfrom a two-component superlattice composed of magnetized ferrite and semiconductor layers.To derive the constitutive parameters of the gyrotropic medium, a homogenization procedurefrom the effective medium theory is applied. The study is carried out in the frequency rangenear the frequency of ferromagnetic resonance, where the magnetic subsystem possesses theproperties of natural hyperbolic dispersion. The topological transitions from an open type-Ihyperboloid to several intricate hyperbolic-like forms are demonstrated for the extraordi-nary waves. We reveal how realistic material losses change the form of isofrequency surfacecharacterizing hyperbolic dispersion. The obtained results broaden our knowledge on thepossible topologies of isofrequency surfaces that can appear in gyrotropic media influencedby an external static magnetic field.

I. INTRODUCTION

Hyperbolic metamaterials can serve as a novel func-tional platform for waveguiding, imaging, sensing, quan-tum and thermal engineering beyond conventional de-vices. These metamaterials derive their name from thehyperbolic topology of isofrequency surfaces characteriz-ing dispersion features of the medium. Hyperbolic meta-materials utilize the concept of engineering the basic dis-persion relations of waves to provide unique electromag-netic modes that can have a broad range of applications(see a comprehensive review on the theory and applica-tion of hyperbolic metamaterials in Refs. 1–6).

A hyperbolic metamaterial is associated with an ex-tremely anisotropic medium in which the principal com-ponents of permittivity or permeability tensor have dif-ferent signs. In the ideal case, in the absence of dissi-pation and spatial dispersion, the isofrequency surfaceof the Fresnel’s equation for such a medium acquiresan open form of the one-fold or two-fold hyperboloid ofrevolution.7 This form of the dispersion topology differsdrastically from the usual case of closed elliptical disper-sion typical for conventional anisotropic media.8 Metal-lic wire composites9 and planar metal-dielectric finely-stratified structures (superlattices)10,11 are two commondesigns of metamaterials demonstrating hyperbolic dis-persion. The optical properties of such artificial struc-tures can be described in the framework of the effectivemedium theory (EMT).12

The most important property of hyperbolic metama-terials is related to the behavior of waves with largemagnitude wavevectors (high-k waves). In vacuum, suchhigh-k waves are evanescent modes which decay exponen-

a)Electronic mail: tvr@jlu.edu.cnb)Electronic mail: volodymyr.i.fesenko@gmail.com

tially. However, in hyperbolic media the open form of theisofrequency surface allows for propagating waves withlarge wavevectors.13 The possibility of propagating high-k waves leads to several unusual effects, including strongenhancement of spontaneous emission,14,15 broadbandinfinite density of states,16 super-resolution imaging,17,18

and abnormal scattering.19

The ability to tune and switch the properties of hyper-bolic metamaterials can greatly broaden their applicationin many fields.20 In particular, in hyperbolic metamate-rials, it is important to gain control over the topologicaltransitions, which may allow performing switching be-tween hyperbolic and elliptic dispersion states. The moststraightforward way to realize such a control is to usemagneto-active components in the metamaterial compo-sition and an external static magnetic field as a drivingagent.21–26

However, the application of an external static mag-netic field modifies the property of the medium, whichbecomes gyrotropic. A gyrotropic medium is charac-terized by permittivity or permeability tensor havingantisymmetric off-diagonal parts. Gyrotropy changesthe dispersion characteristics of the medium and causesthe nonlocality27 and nonreciprocity effects.28 Moreover,magneto-optic materials can exhibit natural hyperbolicdispersion. In particular, in gyromagnetic materials(e.g., ferrites), the hyperbolic dispersion originates fromthe ferromagnetic resonance,29,30 whereas in gyroelectricmaterials (e.g., semiconductors), it appears due to theplasma resonance.31 Therefore, the simultaneous pres-ence of natural and engineered hyperbolic dispersion ina complex structure can lead to the appearance of veryintricate forms of isofrequency surfaces32,33 (see Refs. 34and 35 for a complete taxonomy of isofrequency surfacesthat can be realized in uniaxial anisotropic and bian-isotropic optical materials, respectively).

It is obvious that combining gyroelectric and gyromag-netic materials into a single gyroelectromagnetic system

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Magnetically induced topological transitions 2

can bring many unique dispersion features, which areunattainable in separate subsystems.36–43 In particular,in the present paper, we demonstrate that in a biax-ial gyrotropic medium composed of magnetized ferriteand semiconductor layers, some specific distortions ofisofrequency surfaces may occur. These distortions man-ifest themselves near the frequency of ferromagnetic reso-nance, where the magnetic subsystem possesses the prop-erties of natural hyperbolic dispersion. We first study theidealized lossless structure and then analyze the effect ofactual losses on the hyperbolic dispersion. Thus, we showthat in composite structures containing magneto-activecomponents, in addition to the possibility of control, onecan obtain a diversity of open hyperbolic-like topologiesof isofrequency surfaces.

II. DISPERSION EQUATION

In what follows, we study characteristics of plane elec-tromagnetic waves (bulk waves44) propagating in an ar-bitrary direction through an infinite homogeneous gy-rotropic medium. The gyrotropic medium is magnetizedup to the saturation level in the presence of a uniform

magnetic field ~M . We introduce the Cartesian coordi-nate system with the z axis directed along the vector~M . In the chosen framework, the magnetized gyrotropic

medium can be characterized by the second-rank tensorsof permittivity and permeability

ε =

εxx εxy 0−εxy εyy 0

0 0 εzz

, µ =

µxx µxy 0−µxy µyy 0

0 0 µzz

, (1)

which establish the relations between the electric andmagnetic fields and inductions as(

~D~B

)=

(ε 00 µ

)(~E~H

). (2)

Time and space harmonic variations of the electric ( ~E)

and magnetic ( ~H) components of the wave are given by

~E( ~H) = ~E0( ~H0) exp [i(−ωt+ kxx+ kyy + kzz)] , (3)

where kx, ky, and kz are projections of the wavevector~k in the Cartesian coordinates, and ω is the angular fre-quency.

Starting from a pair of the curl Maxwells equations

∇ × ~E = ik0 ~B and ∇ × ~H = −ik0 ~D in the absence ofsources in the volume of medium (∇· ~D = 0 and ∇· ~B =0), one can derive a system of two coupled wave equationsfor the z-components of the electromagnetic field:45(

ξζ + ζk2zεzz − k20ξεzzµ⊥ k0kzµzz (χ− kxkyη)−k0kzεzz (χ− kxkyη) ξζ + ξk2zµzz − k20ζµzzε⊥

)×(Ez

Hz

)= 0, (4)

where k0 = ω/c, c is the speed of light in vacuum, η =εxxµyy − εyyµxx, ξ = k2xεxx + k2yεyy, ζ = k2xµxx + k2yµyy,

χ = ζεxy + ξµxy, and ε⊥ = εxxεyy + ε2xy and µ⊥ =

µxxµyy+µ2xy are two generalized transverse effective con-

stitutive parameters of the gyrotropic medium.The nontrivial solution of system (4) exists when its de-

terminant of coefficients equals to zero. Under this con-dition, the biquadratic equation describing propagationof bulk waves through the gyrotropic medium is obtainedin the form:32,33

(εzzµzz)−1k4xεxxµxx + k4yεyyµyy + k4zεzzµzz + k2xk

2y

× (εxxµyy + εyyµxx) + k2xk2z(εxxµzz + εzzµxx) + k2yk

2z

× (εyyµzz + εzzµyy)− k20[k2x(εxxεzzµ⊥ + µxxµzzε⊥)

+ k2y(εyyεzzµ⊥ + µyyµzzε⊥) + k2zεzzµzz(εxxµyy

+ εyyµxx − 2εxyµxy)]

+ k40ε⊥µ⊥ = 0. (5)

This equation is also known as the Fresnel’s equation forwave normals.8 It implicitly determines the dispersion re-lation, i.e., the frequency as a function of the wavevector.At a fixed frequency ω, it defines the wavevector surface(isofrequency contour) given in the k-space.

Using the spherical coordinate system in which kx =k sin θ cosϕ, ky = k sin θ sinϕ, and kz = k cos θ, Eq. (5)can be compactly rewritten in therms of κ = k/k0 as

Aκ4 + Bκ2 + C = 0, (6)

where A = (εzzµzz)−1(ε sin2 θ + εzz cos2 θ)(µ sin2 θ +µzz cos2 θ), B = −[(εxxµyy + µxxεyy − 2εxyµxy) cos2 θ +

(εzzµzz)−1(ε⊥µµzz + µ⊥εεzz) sin2 θ], C = ε⊥µ⊥, ε =εxx cos2 ϕ+ εyy sin2 ϕ, and µ = µxx cos2 ϕ+ µyy sin2 ϕ.

Solution of Eq. (6) is straightforward:

κ = ±

√B±√B2 − 4AC

2A. (7)

When the dispersion characteristics of the componentsof tensors (1) are known, solution (7) can be used tocalculate the isofrequency surfaces for the bulk wavespropagating through the infinite homogeneous gyrotropicmedium.

III. HOMOGENIZATION CONDITIONS

In order to realize a gyrotropic medium, we consider arealistic finely-stratified structure made in the form of atwo-component superlattice. In particular, such a struc-ture can be realized for operating in the microwave partof spectrum.46–48 It is composed of ferrite (with constitu-tive parameters εm, µm) and semiconductor (with con-stitutive parameters εs, µs) layers with thicknesses dmand ds, respectively. The layers are periodically arrangedalong the y axis. The structure’s period is L = dm + ds,and the number of periods in the superlattice is largeenough that it can be considered infinite. We supposethat all dimensions dm, ds, and L are much smallerthan the wavelength in the corresponding layer dm λ,ds λ, and period L λ of the superlattice (the long-wavelength limit). Both ferrite and semiconductor sub-systems are magnetized uniformly by an external static

Magnetically induced topological transitions 3

m s

k

gyrotropic

medium

two-component

superlattice

ε μ

θ

d d

z

yx

Mm mε μ

s sε μ

z

yx

φ

FIG. 1. The problem sketch related to both a two-componentsuperlattice influenced by an external static magnetic field ~Mand resulting homogenized gyrotropic medium.

magnetic field ~M directed along the z axis transverselyto the structure periodicity, as shown in Fig. 1.

Taking into account the long-wavelength limit, thefinely-stratified structure can be equivalently representedby a homogeneous anisotropic medium.49 In such a quasi-static approximation, the homogenization procedure50

from the effective medium theory is applied to derivecomponents of tensors (1) in an explicit form. They canbe obtained by substituting µ → G and ε → G to thetensor51,52

G =

Gxx Gxy 0−Gxy Gyy 0

0 0 Gzz

(8)

with components Gxx = g(m)xx δm + g

(s)xx δs + (g

(m)xy −

g(s)xy )2δmδsD, Gxy = (g

(m)xy g

(s)yy δm + g

(s)xy g

(m)yy δs)D, Gyy =

g(m)yy g

(s)yy D, Gzz = g

(m)zz δm+g

(s)zz δs, δm = dm/L, δs = ds/L,

δm + δs = 1, D = (g(s)yy δf + g

(m)yy δs)

−1, where the ex-

pressions for parameters of the tensors g(j) are given inRefs. 31, 45, and 53, and Appendix A. These expres-sions take into account the resonant nature of disper-sion characteristics of the constitutive parameters of fer-rite and semiconductor subsystems caused by the appliedexternal static magnetic field. To verify the applicabil-ity of our approximation, the results of the homogeniza-tion procedure are checked against those of the rigoroustransfer-matrix technique,54,55 for which a good correla-tion is found.

For the homogenized medium, all principal compo-nents of tensors (1) are different (Gxx 6= Gyy 6= Gzz), andthe off-diagonal component is a nonzero value (Gxy 6= 0).This corresponds to the conditions of a biaxial gyrotropiccrystal.56 Two axes of anisotropy are conditioned by thesimultaneous effect of both structure periodicity and ex-ternal static magnetic field influence. The presence oftwo axes of anisotropy supplemented by gyrotropy leadsto several peculiarities in the dispersion characteristicsof the medium. Especially, these peculiarities are mostpronounced in the frequency range near the frequencyof ferromagnetic resonance. In this range the principalcomponents of the permeability tensor vary from positiveto negative values which results in the distinctive topo-logical transitions of the isofrequency surfaces, which arethe subject of our following numerical investigation.

IV. TOPOLOGICAL TRANSITIONS

A. Lossless case

When the direction of the wavevector ~k is given, Eq. (6)is a quadratic equation for κ2. In general, Eq. (6) hastwo different roots. For a lossless medium, each roottakes either a purely imaginary or a purely real value.The real roots correspond to propagating waves. There-fore, two bulk waves with different wavevectors can bepropagated in any direction in the medium.8 Under theaccepted notations,57 the root in solution (7) with up-per sign ‘+’ is attributed to the ‘ordinary’ waves, whileanother root with lower sign ‘−’ is related to the ‘extraor-dinary’ waves. In our subsequent consideration, we aremainly interested in the propagation features of extraor-dinary waves since only these waves demonstrate unusualtopological transitions of their isofrequency surface in thechosen frequency range.

The topological form of isofrequency surface dependson the signs of the principal components of tensors (1)characterizing the medium. For the biaxial gyrotropicmedium under study, the principal components εxx andεyy are positive quantities and µzz = 1 in the entire cho-sen frequency range. Thus, the dispersion topology ismostly conditioned by values and signs of the remain-ing three principal components of tensors (1). Amongthese components, εzz is always negative quantity whileµxx and µyy demonstrate resonant behaviors by chang-ing sign. The dispersion features of all principal compo-nents of tensors (1) are presented in Fig. 2(a). In thisfigure, we distinguish regions of different combinations ofpositive and negative values of µxx and µyy by differentcolors and symbols. They are: Region 1 where (µxx >0) ∧ (µyy > 0), Region 2 where (µxx < 0) ∧ (µyy < 0),and Region 3 where (µxx < 0)∧ (µyy > 0). In addition,the transition points on the k0 scale where componentsµxx and µyy change sign are marked by arrows. Thepoint at k0 ≈ 148.12 m−1 corresponds to the frequencyof ferromagnetic resonance in the ferrite subsystem.

The corresponding three-dimensional (3D) dispersionrelationships of the extraordinary bulk waves propagat-ing through the biaxial gyrotropic medium are plotted inFigs. 2(b) and 2(c). They are supplemented by severaltwo-dimensional (2D) isofrequency contours, which aredrawn at the bottom of these plots. One can see thatthe topological form of wavevector surface changes dras-tically during the transition from one region to another.In each region the isofrequency contours are different forthe kx−kz and ky−kz planes since the medium has twoaxes of anisotropy. Nevertheless, in the entire frequencyrange, the wavevector surface appears in an open hyper-bolic form imposed by the extremely anisotropic tensorε having a single negative component (εzz).

All representative topological forms that arise in thechosen frequency range are collected in Fig. 3. In partic-ular, in Region 1 the isofrequency surface appears as atwo-fold (Type I) hyperboloid oriented along the z axis[Fig. 3(a)]. This form of topology is well known and typi-cal for hyperbolic metamaterials.7 When approaching thefrequency of ferromagnetic resonance, the isofrequency

Magnetically induced topological transitions 4

25 -25

140

148

156

0-1

yy

0.5

148.12

(a) (b) (c)

141414222

142

141488

148

151555

140

148

156

0

0

0

0

140

148

156

20 20

20 20

-20-20

40 40141414222

142

151515151555

141488

148

149

149914149

-1 -1

151.15

1

2

3

yyyyy

4.5 2.5 0

yyyyyyy

yy

k ,

FIG. 2. (a) Dispersion features of principal components of tensors (1) characterizing gyrotropic medium obtained from homoge-nization of two-component superlattice with parameters δm = 0.06 and δs = 0.94, and corresponding 3D dispersion relationshipsand 2D isofrequency contours plotted in (b) kx − kz and (c) ky − kz planes. All material parameters of the superlattice arederived from Eq. (A1) for the saturation magnetization of 2930 G. They are: ω0/2π = 4.2 GHz, ωm/2π = 8.2 GHz, εm = 5.5(for ferrite layers), and ωp/2π = 10.5 GHz, ωc/2π = 9.5 GHz, εl = 1.0, µs = 1.0 (for semiconductor layers).

surface becomes a slightly deformed two-fold hyperboloid[Fig. 3(b)]. Such a distortion was previously observed forchiral58 and gyromagnetic metamaterials,59 although inour case the difference is that the axial symmetry of thehyperboloid is violated by magnetization. When passingthrough the frequency of ferromagnetic resonance, thesurface distortion increases. Specifically, in Region 2

the isofrequency surface transits to the form of cones cutinto either two or four parts which are oriented alongthe z axis, as shown in Figs. 3(c)-3(e). These changesare caused by the extremely anisotropic tensors ε andµ, where either one or two components become negative.To the best of our knowledge, such forms of isofrequencysurface have not previously been encountered in the liter-ature. Finally, in Region 3 , the isofrequency surface ap-pears in the form of two one-fold (Type II) hyperboloidswith orthogonal revolution axes [Fig. 3(f)]. This formof isofrequency surface is attributed to the bi-hyperbolictopology reported recently.32,33

Analysis of parameters included in solution (7) allowsus to determine the areas of existence (continua) of ex-traordinary waves. In fact, these areas are determinedonly by the parameter A which is substituted in the de-nominator of Eq. (7). To confirm this, in Fig. 4 we plotthe dependence of the parameter A on the angles θ and ϕsince these angles determine the direction of wave prop-agation in space. The calculations are carried out attwo fixed values of k0 which belong to Region 2 . Theparametric surfaces of A are supplemented by plots ofcorresponding values of the real root κ of Eq. (6). Atthe bottom of each plot, the contours of surfaces A andκ are drawn, where the continua of extraordinary wavesare filled with blue.

From a comparison of data plotted in Figs. 3(d) and4(a) as well as in Figs. 3(e) and 4(b), one can concludethat the number of continua of extraordinary waves andtheir position on the θ−ϕ plane is determined by the ex-trema of the parameter A. In particular, in Region 2 ex-traordinary waves exist where A < 0, and their continua

are centered around the local minima of the parameter A.The condition A = 0 determines the transition between

the propagating and non-propagating states of extraordi-nary waves. From this condition, a set of two equationsfollows

(εxx cos2 ϕ+ εyy sin2 ϕ) sin2 θ + εzz cos2 θ = 0, (9)

(µxx cos2 ϕ+ µyy sin2 ϕ) sin2 θ + µzz cos2 θ = 0, (10)

which outlines the boundaries of continua of extraordi-nary waves projected onto the θ−ϕ plane. For the givenfrequency and parameters of the superlattice, the equal-ity εxx ≈ εyy holds [see Fig. 2(a)]. This equality sug-gests that Eq. (9) does not depend on ϕ. To illustratethis, the solutions of Eqs. (9) and (10) are presented inFig. 4(c). One can conclude that the constitutive pa-rameters of the semiconductor subsystem determine thepropagation conditions of the wave along a certain po-lar angle θ, while those of the ferrite subsystem are re-sponsible for the continua variations with the azimuthalangle ϕ.

B. Impact of losses

In the vicinity of the frequency of ferromagnetic res-onance, the material losses in ferrite become significantand thus should be accounted for. In hyperbolic meta-materials, the presence of losses leads to a change in theform of isofrequency surfaces, which can undergo signif-icant loss-induced modifications.33,60,61 Therefore, it isimportant to study how losses affect dispersion character-istics of bulk waves propagating through the given biax-ial gyrotropic medium. As before, our main interest is inrevealing the conditions of the extraordinary waves prop-agation in the frequency range belonging to Region 2 .

In the presence of losses, the components of tensors(1) become complex quantities εij = ε′ij + iε′′ij and µij =

Magnetically induced topological transitions 5

30

30

30 -30

-30

-30

(a) (b)

(f)

10

10

10 -10

-10

-10

20

20

20 -20

-20

-20

-1= 1480

-1= 1420

-1= 1550

50

50

50 -50

-50

-50

(c) (d)

50

50

50 -50

-50

-50

-1= 1490

-1= 148.50

(e)

50

50

50 -50

-50

-50

-1= 149.50

FIG. 3. Topological forms of isofrequency surface related to extraordinary bulk waves propagating through an idealized (lossless)biaxial gyrotropic medium. They correspond to (a), (b) Region 1©, (c)-(e) Region 2©, and (f) Region 3©. All parameters of thesuperlattice are the same as in Fig. 2.

µ′ij + iµ′′ij (i, j = x, y, z), which yields that the wavevec-

tor ~k has complex-valued components kj = k′j + ik′′j(j = x, y, z). In general, all four roots of Eq. (6) arecomplex values κj = κ′j + iκ′′j (j = 1, 2, 3, 4), and amongthem, only two solutions are physically acceptable. Aphysically acceptable solution implies that the wave de-cays as it propagates through a lossy medium. In accor-dance with the known notations,62,63 two physically ac-ceptable roots of Eq. (6) correspond to the propagationconditions of the ‘proper’ complex modes, while the tworemaining roots are related to the ‘improper’ complexmodes. In particular, for the chosen time dependence

factor exp(−iωt), the wave with κ′′j > 0 satisfies the phys-ically acceptable propagation condition and is related tothe proper complex modes, while the wave with κ′′j < 0has exponentially growing amplitude during propagationand belongs to the improper complex modes. One of thepair of corresponding modes corresponds to the properor improper extraordinary waves.

In our model of the superlattice, the losses are intro-duced via the parameters β and ν for ferrite and semi-conductor layers, respectively (see Appendix A). To il-lustrate the impact of losses on the propagation condi-tions of extraordinary waves, the corresponding roots of

Magnetically induced topological transitions 6

Eq.(9); Eq.(10)(a) (b)

-90

90-180

180

50

25

θ, degφ, deg

κ

0

0 0

-90

90-180

180

50

25

κ

θ, degφ, deg

00

-180

180

-180

180

φ, deg

φ, deg

(c)

90-90

θ, deg

k = 149.0 m0

k = 149.5 m0

-1

-1

FIG. 4. Dependence of values of coefficient A (color maps) and root κ (blue plots) of dispersion equation (6) on the propagationdirection of extraordinary waves when the frequency parameter is fixed at (a) k0 = 149.0 m−1 and (b) k0 = 149.5 m−1. Contoursat the bottom are just projections given for an illustrative purpose. (c) A set of solutions of Eqs. (9) and (10) which outlinesthe corresponding continua of extraordinary waves (blue colored regions) projected on the θ − ϕ plane. All parameters of thesuperlattice are the same as in Fig. 2.

30

50

30

-50

-30 -30

30

50

30

-50

-30 -30

50

30

30 -30

-30

-50

0

00

50

30

30 -30

-30

-50

0

00

(a)

(b)

FIG. 5. Isofrequency surfaces (real parts) and their cross-sections related to propagation conditions of proper extraordinarywaves for a lossy biaxial gyrotropic medium. Parameters of losses in the ferrite and semiconductor subsystems are: (a) β =1 × 10−4, ν = 2 × 10−2 GHz and (b) β = 1 × 10−3, ν = 5 × 10−2 GHz. All other parameters of the superlattice are the sameas in Fig. 2. The frequency parameter is fixed at k0 = 149.0 m−1.

Eq. (6) are selected, and isofrequency surfaces associated with the real part of ~k are plotted in Fig. 5 for two dif-

Magnetically induced topological transitions 7

ferent sets of parameters of losses at the fixed frequency.These parameters of losses are typical for actual ferriteand semiconductor materials. Each plot is supplementedby the cross-sections drawn in the k′x − k′z and k′y − k′zplanes.

One can compare the forms of isofrequency surfacesobtained for the lossy medium with that plotted for thelossless case [see Fig. 3(d)]. From this comparison aconclusion can be made that the hyperbolic topology ofthe isofrequency surfaces of extraordinary waves surviveswhen the losses are introduced into the system. However,their form appears to be sufficiently distorted.

Figure 5 suggests that, as soon as a small amount oflosses is introduced into either ferrite or semiconductorsubsystem of the superlattice, the isofrequency surfaceof extraordinary waves no longer extends to infinity, buttransits to a closed finite form. This transition is in fullcompliance with the results reported earlier for the non-magnetic lossy hyperbolic metamaterials.60,61,64 As fol-lows from changing in the forms of isofrequency surfacesplotted in Figs. 5(a) and 5(b), a gradual increase in lossesreduces the size of the finite existence areas. Thus, whenimplementing magnetic control in hyperbolic materials,presence of losses in the constitutive components cannotbe ignored, since they strongly affect the forms of isofre-quency surfaces for actual structures.

V. CONCLUSIONS

In conclusion, magnetically induced topological tran-sitions of isofrequency surfaces of bulk waves propagat-ing through an unbounded biaxial gyrotropic medium arestudied. The gyrotropic medium is realized from a long-wavelength consideration of a two-component superlat-tice containing ferrite and semiconductor layers. All ourstudies were performed in the frequency range near thefrequency of ferromagnetic resonance.

In the lossless case, the topological transitions of isofre-quency surfaces of extraordinary high-k waves from anopen type-I hyperboloid to the form of a cone cut intoeither two or four parts as well as a bi-hyperboloid aredemonstrated. Such intricate forms of topology are possi-ble when principal components of both permittivity andpermeability tensors become negative quantities. Forsuch an extremely anisotropic medium, the areas of ex-istence (continua) of extraordinary waves are defined,and the conditions which determine the transition be-tween the propagating and non-propagating states ofthese waves are revealed. The forms of isofrequency sur-faces obtained for the lossless case broaden our knowl-edge on the possible topologies of dispersion existing inanisotropic materials.

The presence of losses leads to a strong distortion ofhyperbolic isofrequency surfaces as compared with thelossless case. Therefore, when gaining magnetic controlin hyperbolic metamaterials, in addition to realizing thedesired topology of isofrequency surfaces, it is also nec-essary to analyze obtained solution on its physical feasi-bility, as soon as a system with realistic losses is underconsideration.

Appendix A: Constitutive parameters of ferrite andsemiconductor layers

The expressions for tensors components of the under-lying constitutive parameters of magnetic µm → g(m)

and semiconductor εs → g(s) layers can be written in theform:

g(j) =

g1 ig2 0−ig2 g1 0

0 0 g3

. (A1)

For magnetic layers45,53 the components of tensor g(m)

are: g1 = 1 + χ′ + iχ′′, g2 = Ω′ + iΩ′′, g3 = 1,and χ′ = ω0ωm[ω2

0−ω2(1−β2)]D−1, χ′′ = ωωmβ[ω20 +

ω2(1+β2)]D−1, Ω′ = ωωm[ω20−ω2(1+β2)]D−1, Ω′′ =

2ω2ω0ωmβD−1, D = [ω2

0 − ω2(1 + β2)]2 + 4ω20ω

2β2,where ω0 is the Larmor frequency and β is the dimen-sionless damping constant.

For semiconductor layers31 the components of tensorg(s) are: g1 = εl[1− ω2

p(ω + iν)[ω((ω + iν)2 − ω2c )]−1],

g2 = εlω2pωc[ω((ω + iν)2 − ω2

c )]−1, g3 = εl[1 − ω2p[ω(ω +

iν)]−1], where εl is the part of permittivity of the lattice,ωp is the plasma frequency, ωc is the cyclotron frequencyand ν is the electron collision frequency in plasma.

Relative permittivity εm of the ferrite layers as well asrelative permeability µs of the semiconductor layers arescalar quantities.

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