Quadraxial metamaterial

D. Sakhno,1 E. Koreshin,1 and P. Belov1, *

1ITMO University, Kronverksky pr. 49, 197101 St. Petersburg, Russia

We study the dispersion of electromagnetic waves in a spatially dispersive metamaterial withLorentz-like dependence of principal permittivity tensor components on the respective componentsof the wave vector performing the analysis of isofrequency contours. The considered permittivitytensor describes triple non-connected wire medium. It is demonstrated that the metamaterial hasfour optic axes in the frequency range below artificial plasma frequency. The directions of the opticalaxes do not depend on frequency and coincide with the diagonals of quadrants. The metamaterialsupports two propagating electromagnetic waves in all directions of space except the directions ofaxes. The conical refraction effect is observed for all four optic axes both below and above artificialplasma frequency where the metamaterial supports five propagating waves in most of the directions.

In the general case, any homogeneous local dielectricmedium can be described by symmetric effective permit-tivity tensor which can be diagonalized in some coor-dinate system (in the absence of spatial dispersion andbackground constant magnetic field [1, 2]). The princi-pal elements 𝜀𝑖𝑖 (𝑖 = 𝑥, 𝑦, 𝑧) of the diagonal tensor arecalled principal permittivities [3] and the relations be-tween them determine the shape of dispersion surfaces ofthe medium. There are three types of local dielectric me-dia (𝜀𝑖𝑖 > 0): 1) isotropic media (𝜀𝑥𝑥 = 𝜀𝑦𝑦 = 𝜀𝑧𝑧) withisofrequency surfaces in the form of a sphere; 2) uniaxialmedia (𝜀𝑖𝑖 ̸= 𝜀𝑗𝑗 = 𝜀𝑙𝑙, for some 𝑖 ̸= 𝑗 ̸= 𝑙) with sphericalisofrequency surface for ordinary waves and ellipsoid ofrevolution isofrequency surface for extraordinary waveswhich touch each other in direction of optic axis directedalong 𝑖-axis; 3) biaxial media (𝜀𝑥𝑥 ̸= 𝜀𝑦𝑦 ̸= 𝜀𝑧𝑧) with adoubled sheeted isofrequency surface of the 4th order withtwo optic axes in the plane 𝑖𝑙 such that 𝜀𝑖𝑖 > 𝜀𝑗𝑗 > 𝜀𝑙𝑙[1, 4]. The term of optic axis here is used for direc-tion in which the phase velocities of all electromagneticwaves supported by the material are equal. In biaxialmedia the optical axes feature the effect of conical re-fraction which finds many applications in optics [5]. Theconical refraction appears due to the conical singularityof the isofrequency contour. If not all principal permit-tivities of an uniaxial medium are positive then the ex-traordinary waves in the medium have hyperbolic isofre-quency contours and such medium is called hyperbolicmetamaterial [6]. The biaxial hyperbolic media with𝜀𝑖𝑖 < 0, 0 < 𝜀𝑗𝑗 ̸= 𝜀𝑙𝑙, for some 𝑖 ̸= 𝑗 ̸= 𝑙 are studiedin [7]. Note, that since permittivity tensor elements de-pend on frequency, for some types of biaxial crystals thedirections of main axes depend on the frequency. Thiseffect is called the dispersion of the axes [3].

In the classification presented above the media are as-sumed to be local and no effects of spatial dispersion aretaken into account. In the case of spatially dispersivematerials, the nonlocal effects in the media may stronglyinfluence its dispersion properties. For example, in thecrystals with cubic symmetry the spatial dispersion isdestroying isotropy and leads to the effect called spatial-dispersion-induced birefringence [8–12]. The shape of

dispersion surfaces in nonlocal media is governed by thedependence of permittivity tensor 𝜀(𝜔,k) on wave vector.In this Letter we study dispersion properties of a

medium with strong spatial dispersion described by thepermittivity tensor 𝜀(𝜔,k) of the following form:

𝜀𝑖𝑖(𝜔,k) = 1−𝑘2𝑝

𝑘20 − 𝑘2𝑖, 𝑖 = 𝑥, 𝑦, 𝑧 (1)

where 𝜔 is a frequency, k = (𝑘𝑥, 𝑘𝑦, 𝑘𝑧)T is a wave vector

in the medium, 𝑘0 = 𝜔/𝑐 is a wave number in the freespace, 𝑐 is a speed of light in the free space, 𝑘𝑝 = 𝜔𝑝/𝑐is a wave number corresponding to the artificial plasmafrequency 𝜔𝑝 of the wire medium.

FIG. 1. A unit cell of the triple non-connected wire medium.

The permittivity tensor of the form (1) describes elec-tromagnetic properties of a metamaterial called triplenon-connected wire medium as it was shown in [13–16].The metamaterial consists of three orthogonal to eachother and equally spaced two-dimensional arrays of par-allel infinite straight metal wires (along 𝑥−, 𝑦−, and 𝑧−axes, respectively) located in a free space. In each of thearrays, the axes of the wires form a square lattice in aplane perpendicular to their direction. The wires havethe same radii 𝑟0, the period in all directions is equal to𝑎, and the distances between axes of nearest perpendicu-lar wires is equal to half of the period 𝑎/2. The unit cell

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of the metamaterial is shown in Fig. 1. Mathematically,the geometry can be described by defining coordinates ofthe wire axes in the following way:

(i) the 𝑥-directed wires: 𝑦 = 𝑎𝑛+𝑎/2 and 𝑧 = 𝑎𝑙+𝑎/2,(ii) the 𝑦-directed wires: 𝑥 = 𝑎𝑚+ 𝑎/2 and 𝑧 = 𝑎𝑙,(iii) the 𝑧-directed wires: 𝑥 = 𝑎𝑚 and 𝑦 = 𝑎𝑛,The artificial plasma frequency of the metamaterial

can be determined via its geometrical parameters usingthe following expression [14, 16]:

𝑘2𝑝 =2𝜋/𝑎2

ln(𝑎/2𝜋𝑟0) + 𝜋/6(2)

The effective medium model Eq. (1) correctly de-scribes the properties of wire metamaterial in the quasi-static case when 𝑘𝑖𝑎 << 𝜋 and 𝑘0𝑎 << 𝜋.An eigenmode of the metamaterial with an electric field

in the form E(𝑟) = E𝑒𝑖kr satisfies source-free Maxwellequations:

k×D = 𝜇0𝜔H, k×E = −𝜔D. (3)

Together with the material relation D = 𝜀0=𝜀 E by ex-

cluding magnetic field the following equation can be ob-tained:

𝑘20=𝜀 E =

[︀𝑘2E− (E · k)k

]︀, (4)

where 𝑘0 is a wave number in host media (vacuum) andk – wave vector inside material. Equation (4) can bewritten as a system:⎧⎪⎨⎪⎩

(︀𝜀𝑥𝑥𝑘

20 − 𝑘2𝑦 − 𝑘2𝑧

)︀𝐸𝑥 + 𝑘𝑥𝑘𝑦𝐸𝑦 + 𝑘𝑥𝑘𝑧𝐸𝑧 = 0

𝑘𝑥𝑘𝑦𝐸𝑥 +(︀𝜀𝑦𝑦𝑘

20 − 𝑘2𝑥 − 𝑘2𝑧

)︀𝐸𝑦 + 𝑘𝑦𝑘𝑧𝐸𝑧 = 0

𝑘𝑥𝑘𝑧𝐸𝑥 + 𝑘𝑦𝑘𝑧𝐸𝑦 +(︀𝜀𝑧𝑧𝑘

20 − 𝑘2𝑥 − 𝑘2𝑦

)︀𝐸𝑧 = 0

(5)

By equating the determinant of this system of equationsto zero one can obtain the dispersion equation in thefollowing form [3, 14]:(︀𝜀𝑥𝑥𝑘

20 − 𝑘2𝑦 − 𝑘2𝑧

)︀ (︀𝜀𝑦𝑦𝑘

20 − 𝑘2𝑥 − 𝑘2𝑧

)︀ (︀𝜀𝑧𝑧𝑘

20 − 𝑘2𝑥 − 𝑘2𝑦

)︀−

−(︀𝜀𝑥𝑥𝑘

20 − 𝑘2𝑦 − 𝑘2𝑧

)︀𝑘2𝑦𝑘

2𝑧 −

(︀𝜀𝑦𝑦𝑘

20 − 𝑘2𝑥 − 𝑘2𝑧

)︀𝑘2𝑥𝑘

2𝑧−

−(︀𝜀𝑧𝑧𝑘

20 − 𝑘2𝑥 − 𝑘2𝑦

)︀𝑘2𝑥𝑘

2𝑦 + 2𝑘2𝑥𝑘

2𝑦𝑘

2𝑧 = 0. (6)

Substitution of expressions for components of the

tensor=𝜀 from Eq. (1) into the dispersion equa-

tion Eq. (6) and multiplication of the latter by(︀𝑘20 − 𝑘2𝑥

)︀ (︀𝑘20 − 𝑘2𝑦

)︀ (︀𝑘20 − 𝑘2𝑧

)︀/𝑘20 leads to the dispersion

equation in the form of a polynomial:[︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀3 (︀𝑘40 + 𝑘2𝑥𝑘

2𝑦 + 𝑘2𝑥𝑘

2𝑧 + 𝑘2𝑦𝑘

2𝑧

)︀+

+[︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀ {︀(︀𝑘2𝑥 + 𝑘2𝑦

)︀ (︀𝑘2𝑥 + 𝑘2𝑧

)︀ (︀𝑘2𝑦 + 𝑘2𝑧

)︀×

×(︀𝑘20 − 2𝑘2𝑝 − 𝑘2

)︀− 𝑘4𝑝

(︀𝑘2𝑥𝑘

2𝑦 + 𝑘2𝑥𝑘

2𝑧 + 𝑘2𝑦𝑘

2𝑧

)︀}︀+

+ 2𝑘4𝑝𝑘2𝑥𝑘

2𝑦𝑘

2𝑧 = 0 (7)

FIG. 2. Isofrequency surfaces of metamaterial with permittiv-ity tensor 𝜀(𝜔,k) described by Eq. (1) obtained by numericalsolution of Eq. (7) for frequencies (a) 𝜔 = 0.3𝜔𝑝 and (b)𝜔 = 1.01𝜔𝑝: below and right above the plasma frequency,respectively.

Here we have to stress, that this equation is differentas compared to Eq. (44) from the paper [14]:(︀

𝑘20 − 𝑘2𝑥)︀ (︀

𝑘20 − 𝑘2𝑦)︀ (︀

𝑘20 − 𝑘2𝑧)︀ [︀

𝑘20 − 𝑘2𝑝 − 𝑘2]︀×

×[︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀ [︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀−

−(︀𝑘20 − 𝑘2𝑥

)︀ [︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀𝑘2𝑦𝑘

2𝑧𝑘

4𝑝−

−(︀𝑘20 − 𝑘2𝑦

)︀ [︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀𝑘2𝑥𝑘

2𝑧𝑘

4𝑝−

−(︀𝑘20 − 𝑘2𝑧

)︀ [︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀𝑘2𝑥𝑘

2𝑦𝑘

4𝑝 + 2𝑘2𝑥𝑘

2𝑦𝑘

2𝑧𝑘

6𝑝 = 0

(8)

It can be shown that the Eq. (8) is equal to Eq. (7)multiplied by (𝑘20 − 𝑘2). This means that Eq. (8) hasa spurious solution 𝑘20 − 𝑘2 = 0 which is not a solutionof the original Eq. (6). Thus, in practice it is better to

3

FIG. 3. (a) Isofrequency surface within the first octant for 𝜔 = 0.3𝜔𝑝 and isofrequency contours (b) in 𝑘𝑧 = 0 plane and (c) in

𝑘𝑥 = 𝑘𝑦 plane (𝑘2𝑑 = 𝑘2

𝑥 + 𝑘2𝑦). The conical point in this octant is marked by 𝐷

(+)1 .

use Eq. (7) instead of Eq. (8) in order to avoid spurioussolutions.

The dispersion equation in the form of Eq. (7) al-lows us to study dispersion properties of the metamate-rial via analysis of isofrequency surfaces. Note that inthe literature it is typical to illustrate dispersion proper-ties of triple wire medium by isofrequency contours [16].However such approach limits consideration to behaviorof dispersion in particular plane in 𝑘− space and doesnot reveal full 3D picture of dispersion. That is whyin this Letter we concentrate our attention in particu-lar on isofrequency surfaces and do accompany them byisofrequency contours in symmetry planes. The typicalisofrequency contours obtained by numerical solution oftranscendental Eq. (7) are shown in Fig. 2.

The symmetry of isofrequency surfaces in the recipro-cal space can be analysed according to the symmetry ofthe Eq. (7) itself. There are 9 mirror symmetry planesin the reciprocal space: first three planes are 𝑘𝑥 = 0,𝑘𝑦 = 0 and 𝑘𝑧 = 0 (due to the equation insensitivityto substitutions 𝑘𝑥 → −𝑘𝑥, 𝑘𝑦 → −𝑘𝑦 and 𝑘𝑧 → −𝑘𝑧),the other planes are main diagonals planes 𝑘𝑥 = ±𝑘𝑦,𝑘𝑦 = ±𝑘𝑧, 𝑘𝑧 = ±𝑘𝑥 (due to the equation insensitivity tocorresponding substitutions). At the intersections of theplanes, there are rotational axes of symmetry, which canbe shown from the Eq. (7) via a more complex substi-tutions: three 4-fold rotation axes (𝑘𝑥, 𝑘𝑦 and 𝑘𝑧 axes)and four 3-fold axes (𝑘𝑥 = ±𝑘𝑦 = ±𝑘𝑧).

In the symmetry plane 𝑘𝑧 = 0 Eq. (7) reduces to thefollowing form:

[︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀ {︁ [︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀2 (︀𝑘40 + 𝑘2𝑥𝑘

2𝑦

)︀+

+(︀𝑘20 − 2𝑘2𝑝 − 𝑘2

)︀𝑘2𝑘2𝑥𝑘

2𝑦 − 𝑘4𝑝𝑘

2𝑥𝑘

2𝑦

}︁= 0 (9)

In the diagonal symmetry plane 𝑘𝑥 = 𝑘𝑦 Eq. (7) can

be converted as:[︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀3 (︀𝑘40 + 𝑘4𝑥 + 2𝑘2𝑥𝑘

2𝑧

)︀+

+[︀𝑘20 − 𝑘2𝑝 − 𝑘2

]︀ {︁2𝑘2𝑥

(︀𝑘2𝑥 + 𝑘2𝑧

)︀2 (︀𝑘20 − 2𝑘2𝑝 − 𝑘2

)︀−

− 𝑘4𝑝(︀𝑘4𝑥 + 2𝑘2𝑥𝑘

2𝑧

)︀}︀+ 2𝑘4𝑝𝑘

4𝑥𝑘

2𝑧 = 0 (10)

In Fig. 3 and 4 we plot isofrequency surfaces restrictedto the first octant of reciprocal space accompanied byisofrequency contours in 𝑘𝑧 = 0 and 𝑘𝑥 = 𝑘𝑦 planes (cal-culated through solution of Eqs. (9) and 10) for typicalfrequencies below plasma frequency and above, respec-tively. The polarization of the eigenwaves correspondingto different branches of isofrequency contours is identifiedusing Eq. (5) and marked as either TE or TM in Figs.3(b,c) and 4(b,c). Note, that the behavior of isofrequencycontours is different for frequencies below and above ar-tificial plasma frequency that is why we study these twocases separately in more details.

At low frequencies the isofrequency surface is a singlesheet surface of complex shape with 8 conical points and

6 asymptotic planes 𝑘𝑖 = ±𝑘0. The conical points 𝐷(+)𝑛

(here 𝑛 = 1..8 is a number of an octant) lie at the maindiagonals of the octants and their coordinates can beeasily obtained from solution of Eq. (10):

𝐷(+)𝑛 =

√︂𝑘03

(︁2𝑘0 +

√︁𝑘20 + 3𝑘2𝑝

)︁(±1,±1,±1)𝑇 . (11)

For all directions of propagation except directions Γ𝐷(+)𝑛

the metamaterial supports two waves with two differentwave vectors. Along directions corresponding to the con-

ical points 𝐷(+)𝑛 the metamaterial supports single wave

and thus these directions correspond to optic axes of the

metamaterial. Since 𝐷(+)𝑛 have pairs of points symmetric

with respect to the Γ point the metamaterial has 4 opticaxes corresponding to the main diagonals of octants. Itis important to note that in contrary to biaxial crystalsthe metamaterial does not suffer from dispersion of the

4

FIG. 4. (a) Part of isofrequency surface within the first octant for 𝜔 = 1.01𝜔𝑝 located near Γ = (0, 0, 0)T and isofrequencycontours (b) in 𝑘𝑧 = 0 plane and (c) in 𝑘𝑥 = 𝑘𝑦 plane (𝑘2

𝑑 = 𝑘2𝑥 + 𝑘2

𝑦). The conical points belonging to this part of the

isofrequency surface in this octant are marked by 𝐷(−)1 (the one located at the diagonal of octant) and 𝐴

(+)𝑖 (the ones located

at the 𝑖-th axis).

axes [3] since directions of the axes are fixed by the sym-metry of the metamaterial and does not depend on thefrequency.

At the frequencies above plasma frequency thecomplex-shaped surface with 8 conical points and 6asymptotic planes 𝑘𝑖 = ±𝑘0 described above is accom-panied by an additional three-sheeted surface located invicinity of the origin point (see Fig. 4(a) where enlargedimage of the surface is provided). The three sheet sur-face has 14 conical points: 8 points where two sheetsintersect and 6 points corresponding to intersection of 3

sheets. The 8 conical points 𝐷(−)𝑛 (here 𝑛 = 1..8 is a

number of an octant) lie at the main diagonals of the oc-tants and their coordinates can be easily obtained fromsolution of Eq. (10):

𝐷(−)𝑛 =

√︂𝑘03

(︁2𝑘0 −

√︁𝑘20 + 3𝑘2𝑝

)︁(±1,±1,±1)𝑇 . (12)

The 6 conical points corresponding to intersection of 3sheets lies at the coordinate axes and their coordinatescan be easily found by soling Eq. (9) and taking intoaccount symmetry of the dispersion equation:

𝐴(±)𝑥 =

(︁±√︁

𝑘20 − 𝑘2𝑝, 0, 0)︁

(13)

𝐴(±)𝑦 =

(︁0,±

√︁𝑘20 − 𝑘2𝑝, 0

)︁. (14)

𝐴(±)𝑧 =

(︁0, 0,±

√︁𝑘20 − 𝑘2𝑝

)︁. (15)

Formally, it turns out that the metamaterial has 4optic axes corresponding to diagonals of the octants if𝜔 < 𝜔𝑝 (axes 1-4, see Fig. 2(a)) and 3 optic axes cor-responding to coordinate axes if 𝜔 > 𝜔𝑝 (axes 5-7, see

Fig. 2(b)) . Whereas the number of axes with conicalrefraction is equal 7 if 𝜔 > 𝜔𝑝 (axes 1-7, see Fig. 2(b)).

In order to check the theoretical prediction about num-ber of optic axes of the metamaterial we performed nu-merical simulation of dispersion properties of triple non-connected wire medium with a unit cell shown in Fig.1 with 𝑎 = 10mm and 𝑟0 = 0.05𝑎 using CST Mi-crowave Studio software. The plasma frequency for thismetamaterial was numerically identified to be equal to𝜔𝑝𝑎/2𝜋𝑐 ≈ 0.3058.

The simulation results in Fig. 5(a) presents isofre-quency contours in the diagonal plane 𝑘𝑥 = 𝑘𝑦 (the fourthpart of the diagonal plane of the ΓMR zone section) for𝜔 = 0.327𝜔𝑝. The blue and red arrows show, respectively,the directions of electric and magnetic fields calculatedby averaging local field distributions over the unit cell atcertain points of the contour [17]. One can see very goodagreement between Fig. 5(a) and Fig. 3(c) where sim-ilar isofrequency contours were calculated theoretically.The TE and TM branches of isofrequency contours cross

each other at𝐷(+)1 point and this numerically confirm the

fact that the diagonal is an optic axis of the metamate-rial. Another mode crossing is observed at the boundaryof the Brillouin zone 𝑘𝑧 = 𝜋/𝑎 since under such conditionthe sets of wires oriented along z-axis does not interactwith wires oriented along x- and y-axes as it clearly fol-lows from Eq. (27) of Ref. [14].

Numerically obtained isofrequency contours aroundthe Γ-point above the plasma frequency 𝜔 = 1.014𝜔𝑝

which correspond to 𝑘𝑧 = 0 and 𝑘𝑥 = 𝑘𝑦 planes are shownin Fig. 5(b) and (c), respectively. Comparing the numer-ically obtained contours with the theoretical ones shownin Fig. 4(b,c) one can note a key difference: the predicted

triple self-intersection points 𝐴(±)𝑖 on the main axes (Eq.

(13–15), Fig. 4) split into pairs and no conical refractionis observed. The splitting appears due to the fact that

5

kx /kp

k y/k p

TM

TETM

k z/k p

TM

TE

TM

axis 1

kd /kp

D1(−)

k z/k p

TM

TE axis 1

M

R

Γkd /kp

D1(+)

(a)

(b)

(c)

FIG. 5. Numerically obtained isofrequency contours for thetriple wire metamaterial with polarization designation (blueand red arrows show the direction of E𝑎𝑣 and H𝑎𝑣 respec-tively, averaged over the unit cell) (a) below the plasma fre-quency 𝜔 ≈ 0.327𝜔𝑝 in the 𝑘𝑥 = 𝑘𝑦 plane (𝑘2

𝑑 = 𝑘2𝑥 + 𝑘2

𝑦; Bril-

louin zone coordinates: Γ = (0, 0, 0)T, 𝑀 = (𝜋/𝑎, 𝜋/𝑎, 0)T,𝑅 = (𝜋/𝑎, 𝜋/𝑎, 𝜋/𝑎)T) and and above the plasma frequency𝜔 ≈ 1.014𝜔𝑝 in (b) the 𝑘𝑧 = 0 and (c) the 𝑘𝑥 = 𝑘𝑦 planes(𝑘2

𝑑 = 𝑘2𝑥 + 𝑘2

𝑦).

Eq. (1) does not take into account the transverse polar-ization of wires. At the meantime, the diagonal conical

points 𝐷(−)𝑛 (according to Eq. (12)) remains the same as

in the theoretical case (Fig. 5(c)).

In conclusion, we have demonstrated both theoreti-cally and numerically that the triple non-connected wiremedium has four optic axes (corresponding to the maindiagonals of the quadrants) at frequencies below plasmafrequency. The conical refraction is predicated for all fouroptic axes. The latter effect has a very wide range of ap-

plications, from the creation of optical tweezers to trap-ping of Bose-Einstein condensates [5]. Thus, the triplewire medium is an easy-to-manufacture alternative to bi-axial crystals with adjustable geometry parameters [14]and stable optical axes in a wide frequency range.The authors acknowledge Prof. Maxim Gorlach for

help and fruitful discussions and Nikita Karagodin forhelp with mathematical issues.

6

* denis.sakhno@metalab.ifmo.ru[1] L. D. Landau and E. Lifshitz, Course of Theoretical

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