Optic axis-driven new horizons for hyperbolic metamaterials

Allan D. Boardman1,*, Peter Egan1, and Martin McCall2

1 Joule Physics Laboratory, Materials & Physics Research Centre, University of Salford, Greater Manchester M5 4WT, UK2 Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2AZ, UK

Received 1 September 2015 / Accepted 28 October 2015

Abstract – The broad assertion here is that the current hyperbolic metamaterial world is only partially served byinvestigations that incorporate only some limited version of anisotropy. Even modest deviations of the optic axis fromthe main propagation axis lead to new phase shifts, which not only compete with those created by absorption but endup dominating them. Some progress has been attempted in the literature by introducing the terms ‘‘asymmetric hyper-bolic media’’, but it appears that this kind of asymmetry only involves an optic axis at an angle to the interface of auniaxial crystal. From a device point of view, many new prospects should appear and the outcomes of the investiga-tions presented here yield a new general theory. It is emphasised that the orientation of the optic axis is a significantdeterminant in the resulting optical properties. Whereas for conventional anisotropic waveguides homogeneous prop-agating waves occur over a limited range of angular dispositions of the optic axis it is shown that for a hyperbolicguide a critical angular setting exists, above which the guided waves are always homogeneous. This has significantimplications for metawaveguide designs. The resulting structures are more tolerant to optic axis misalignment.

Key words: Metamaterial, Hyperbolic, Uniaxial, Optic axis.

1 Introduction

The interest in calcite (Iceland spar) dates back to 1669 andthe work of Bartholin [1]. This means that the theory of prop-agation of electromagnetic waves in such uniaxial anisotropiccrystals has naturally been of interest for some considerabletime. Indeed, the early investigations are extremely useful informing a background and vital points of contact for the crea-tion of new horizons in the global surge of interest in what hasbecome known as hyperbolic [2] non-magnetic metamaterial.The current hyperbolic metamaterial world is only partiallyserved by investigations that incorporate some limited versionof anisotropy. In order to develop new pathways, it should firstof all be noted that readily available uniaxial non-metamaterialcrystals, like calcite, have a direction around which rotationalsymmetry is apparent. This direction is called the optic axisand, given a rectangular coordinate system, it is usually setto lie along the z-axis, which in the model presented here isgoing to be the direction of wave propagation. Adopting thisstep leads to the permittivity tensor [3] emerging in diagonalform, possessing equal x and y components that are unequalto the component associated with the z-direction. Hyperbolicmetamaterials have the same x and y tensor propertiesbut possess a negative z-component. A completely general

approach to the constructive manipulation of the optic axisposition in uniaxial crystals is required if a proper descriptionof any wave propagation within them is to be achieved. In thiscontext, as will be emphasized here, there is a need torecognise clearly the dramatic role of anisotropy in uniaxialmetamaterial investigations, as was pointed out some yearsago [4].

In order to set up a full investigation of how the orientationof the optic axis affects wave propagation in hyperbolicmetamaterials, it is efficient to focus upon two broad areas.Area one includes the interrogation of a slab of metamaterial,involving the transmission of light through an interface, or anumber of interfaces. Area two invokes the paramount roleof the optic axis when guided waves are being generated,and it is this case that will be studied here, where it will beshown that for a hyperbolic film, in contrast to the usualnon-metamaterial anisotropic guides, substantial changes inthe character of guided waves occur, presenting the opportunityfor novel device design.

In general, there are off-diagonal permittivity tensor ele-ments to be considered when controlling the optic axis direc-tion. These elements can often be brought in unintentionallyat experimental level when trying to compare measure-ments with simulations. This is emphasised in the work byStarodubtsev [4], in which he writes*e-mail: a.d.boardman@salford.ac.uk

EPJ Appl. Metamat. 2015, 2, 11� A.D. Boardman et al., Published by EDP Sciences, 2016DOI: 10.1051/epjam/2015007

Available online at:http://epjam.edp-open.org

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

OPEN ACCESSRESEARCH ARTICLE

‘‘even small optical anisotropy is a much greater enemy

of the perfect lens than absorption’’.

Thus, even modest deviations of the optic axis from themain z-axis leads to new phase shifts, which not only com-pete with those created by absorption but may end up dom-inating them. Some progress has been attempted byintroducing the term ‘‘asymmetric hyperbolic media’’ [5,6], but it appears that this kind of asymmetry only involvesan optic axis at a certain angle to the interface of a uniaxialcrystal. Here, a full investigation is made of the effect thatthe optic axis direction has on the existence of pure hybridhomogeneous waves in a wave guide consisting of a hyper-bolic film, an asymmetric substrate with all positive permit-tivity tensor elements, and an air cover.

2 The optic axis

The historic, classic work of Knoesen, Gaylord andMoharam [7] on non-metamaterial anisotropic uniaxial wave-guides introduces the geometrical concept of an optic axis inthe context of a slab waveguide. It is an attractive anisotropicdesign, which is why it is adopted here in order to illustrate thenew metamaterial concepts to be introduced in this investiga-tion. It does not limit the generality of the overall conclusionsand, for the moment, all three components of the wave number,related to right angle Cartesian coordinates will be considered.

The classic diagonal form of the permittivity tensor for auniaxial, non-magnetic crystal when the optic axis lies alongthe z-axis is given by

�e ¼e1 0 0

0 e2 0

0 0 e3

0B@

1CA: ð1Þ

Referring to Figure 1, the general direction of the optic axisis described through the angles h and /, where h is the anglethat the optic axis makes with the z-axis, and / is the anglebetween the projection of the optic axis onto the xy-plane mea-sured from the x-axis (Figure 1).

In order to take into account a variation in the orientationof the optic axis, the transformation matrix

T ¼sin / cos h cos / sin h cos /

� cos / sin / cos h sin h sin /

0 � sin h cos h

0B@

1CA

can be used to find the elements of the general permittivitytensor,

e ¼exx exy exz

eyx eyy eyz

ezx ezy ezz

0B@

1CA ð2Þ

in the usual way, using e ¼ T�eT�1. In the lossless cases beinginvestigated, the tensor is Hermitian so that eij ¼ e�ji. Thus, forreal elements, the matrix is symmetric, eij ¼ eji and the ele-ments are given by [8]

exx ¼ e1sin2/þ e1cos2hþ e3sin2hð Þcos2/;

exy ¼ eyx ¼ e3 � e1ð Þ sin / cos /sin2h;

exz ¼ ezx ¼ e3 � e1ð Þ cos / cos h sin h;

eyy ¼ e1cos2/þ e1cos2hþ e3sin2hð Þsin2/

eyz ¼ e3 � e1ð Þ sin / cos h sin h;

ezz ¼ e1sin2hþ e3cos2h ;

; ð3Þ

In special cases, for example when / = 0� and the opticaxis lies in xz-plane [9], or when h = 90� and the optic axislies in the xy-plane [10], the relevant permittivity tensorcan be found using the appropriate values of h and / inequation (3).

For wave propagation, a useful characteristic is the dimen-sionless wave number with components ji ¼ cki

x , where i = x, yor z, c is the velocity of light in free-space and x is the angularfrequency. For the case when all the diagonal tensor elementsare positive, two surfaces in ðjx; jy ; jzÞ space will appear for agiven frequency. One is that of an ellipsoid and the other is thatof a sphere. The former is classically associated with extra-ordinary modes where, in general, the power flow and phasevelocity direction are not parallel. The sphere is associatedwith ordinary modes where the power flow and the phasevelocity directions are the same. For a metamaterial describedby a diagonal dielectric tensor with two equal positive ele-ments, and one negative element, the ordinary wave sphere stillemerges but the ellipsoid opens up into a hyperboloid. Figure 2demonstrates this for the case when the optic axis lies along thez-axis [11].

Following the magnificent lead, set many years ago by therigorous and pioneering general work of Knoesen et al. [7], andsome pioneering work on beam propagation by Fleck and Feit[12], the discussion presented below will focus upon a loss-free, planar, linear uniaxial waveguide (Figure 3), where thesubscripts s, f and c are used to represent the substrate, filmand cover respectively.

In this investigation the planar guide has a hyperbolicmetamaterial central film with e3f < 0, an anisotropic substratewith e3s > 0, and an isotropic cover, assumed for the moment

Figure 1. For a rectangular coordinate system (x, y, z), the generalposition of the optic axis in a uniaxial crystal can be set at an angleh to the z-axis. This axis then has a projection on the (x, y) plane,which is at an angle / to the x-axis.

2 A.D. Boardman et al.: EPJ Appl. Metamat. 2015, 2, 11

to be air. The adoption of losses can be introduced at a laterstage of any device design but it is not a paramount issue heresince hyperbolic metamaterials, unlike double-negative me-tamaterials, are not resonant. The model here has an optic axisthat is permitted to have any orientation, but is the samethroughout for the film, the substrate and the cover [7]. Uncou-pling of the TE and TM modes occurs when the angle / = 0�.Otherwise, the waves are hybrid and will be highlighted interms of pure guided homogeneous modes, pure guided inho-mogeneous modes, or leaky guided modes [7]. A clarificationof the homogeneous modes is that these waves have constantplanar phase fronts, and are totally guided with constant ampli-tudes. Inhomogeneous waves, on the other hand, are character-ised by planar phase fronts with non-constant amplitudes.Leaky modes are not totally guided by a slab waveguide.The main thrust in this paper is in the effect that the orientationof the optic axis has on the existence of pure hybrid homoge-neous modes. The term ‘‘anisotropic’’ will be used when all theelements of the permittivity tensor are positive, and, specifi-cally, when e1 = e2 > 0; e3 > 0 5 e1. The expression ‘‘hyper-bolic’’ will be used to mean anisotropic, with e1 = e2 > 0and e3 < 0.

In the absence of attenuation, the values of jz (the compo-nent of the wavenumber in the propagating direction) are real,and, in the case of an anisotropic film, are restricted by the per-mittivity of the isotropic cover, and the permittivity tensors ofthe anisotropic substrate and the hyperbolic film. In the aniso-tropic film case, the maximum value that jz is can take is

pe1f.

However, under certain orientations of the optic axis, to bedetermined later, this upper boundary no longer applies whenthe film is hyperbolic.

If the optic axis lies in the xz-plane, then the range of val-ues of h over which the important homogeneous waves propa-gate, is restricted to a specific range for both anisotropic andhyperbolic guides. For an anisotropic film, if the optic axis isalso rotated so that angle / has a positive value, then the rangeof h over which homogeneous waves propagate decreases. Incontrast, when using a hyperbolic metamaterial-based guide,increasing / from 0� results in an increase in the range of h

over which pure hybrid homogeneous modes propagate until,at a certain critical value of /, these modes will propagatefor all angles of h between 0� and 90�. At angles of / greaterthan this critical angle, modes are possible with j2

z > e1f butthese modes are hybrid inhomogeneous waves. This will bediscussed in more detail later.

3 Wave propagation and design windows

First of all, consider a bulk non-magnetic uniaxial crystal,with a permittivity tensor eij. This material will support planewaves of the form Aei(k�r-xt), where the complex wave vectoramplitude is A, and wave vector and position vector are,respectively, k ¼ kx; ky ; kz

� �and r ¼ x; y; zð Þ. Maxwell’s

equations then yield the following equation for the electricfield components Ex;Ey ;Ez

� �.

k2 � k2x � x2

c2 exx �kxky � x2

c2 exy �kxkz � x2

c2 exz

�kxky � x2

c2 eyx k2 � k2y � x2

c2 eyy �kykz � x2

c2 eyz

�kxkz � x2

c2 ezx �kykz � x2

c2 eyz k2 � k2z � x2

c2 ezz

0BB@

1CCA

Ex

Ey

Ez

0B@

1CA ¼ 0

ð4Þor, in terms of the dimensionless wave vector ji definedabove

j2y þ j2

z � exx �jxjy � exy �jxjz � exz

�jxjy � exy j2x þ j2

z � eyy �jyjz � eyz

�jxjz � exz �jyjz � eyz j2x þ j2

y � ezz

0B@

1CA

Ex

Ey

Ez

0B@

1CA ¼ 0

ð5Þ

In the planar waveguide (Figure 2), as is usual, it isassumed that there is no variation of the fields in the y-directionso that setting jy ¼ 0 in equation (5) yields the biquadratic

j4xexx þ 2j3

xjzexz þ j2x j2

z exx þ ezzð Þ þ e2xy þ e2

xz � exxezz � exxeyy

h iþ 2jxjz j2

z exz þ exyeyz � exzeyy

� �þ j4

z ezz þ j2z e2

yz þ e2xz � eyyezz � exxezz

� �þ exxeyyezz þ 2exyeyzexz � exxe2

yz � eyye2xz � e2

xyezz ¼ 0

ð6Þ

Figure 2. Cross-sections through the (jx, jz)-plane in wavenumberspace when jy = 0 and the optic axis lies along the jz-axis. Thefigure shows the transition from anisotropic (ellipse) to hyperbolicas e3 changes in incremental steps from a positive to a negativevalue. The element e1 = e2 = 4.0.

Figure 3. Planar, step-index, waveguide with guided waves prop-agating along z, for which the central film (f) is a hyperbolicmetamaterial. The cover (c) and substrate (s) can be purelyisotropic, or anisotropic, with positive tensor elements.

A.D. Boardman et al.: EPJ Appl. Metamat. 2015, 2, 11 3

which, with judicious use of equations (3), reduces to

exxj4x þ 2exzjzj3

x þ j2z exx þ ezzð Þ � e1 exx þ e3ð Þ

� �j2

x

þ 2exzjzjx j2z � e1

� �þ j4

z ezz � j2z e1 ezz þ e3ð Þ þ e2

1e3

which factorises to give

j2z þ j2

x � e1 ¼ 0; exxj2x þ 2jzjxexz þ j2

z ezz � e1e3 ¼ 0: ð7Þ

These equations show how the elliptic case, with only posi-tive diagonal tensor elements, turns into the hyperbolic case ase3 becomes negative (see for example Figure 2). At this stage,it is convenient to introduce the subscripts o and e to label theordinary and extraordinary mode contributions respectively.This step results in the solutions

jox ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffie1 � j2

z

pjex ¼

1

exx�jzexz �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2

z e2xz � exxezz

� �� e1e3exx

q :ð8Þ

For a waveguide, these equations give the jx=jz relation-ship for each layer. Since the interest here is in waves thatdo not attenuate in z-direction, jz must be real. Thus it is clearfrom (8) that for

j2z < e1 and j2

z e2xz � exxezz

� �� e1e3exx > 0 ð9Þ

the solutions for jx are real, and for

j2z > e1 and j2

z e2xz � exxezz

� �þ e1e3exx < 0 ð10Þ

the solution for the ordinary mode will be imaginary, and thesolution for the extraordinary mode will be complex. In theguide, these results will determine whether the modes are

homogeneous, inhomogeneous or leaky modes. For propaga-tion along the guide, fields that decay away from the surfacesinto the semi-infinite media of the substrate and cover aresought. If either or both of these layers are isotropic, then,for guided waves, the first of (10) must be satisfied, and expo-nential decay will occur. Using the second of (10), for an aniso-tropic substrate, the decay will not be exponential due to thecomplex values of the solution for the extraordinary modesgiven by (8). In this case, solutions are sought which determinethat the electric and magnetic fields decay to zero as x! �1.

For the specific cases discussed below it is assumed thatthat the cladding is air and in the anisotropic casee1f > e1s > e3f > e3s > e1c. In the hyperbolic film casee1f > e1s > e3f > e3s > e1c and e3f < 0. Thus, for the constitu-ents of the slab waveguide shown in Figure 3, the necessaryconditions for pure hybrid homogeneous modes are:for the film:

j2z e2

xzf � exxf ezzf

� �þ e1f e3f exxf > 0 and

e1f � j2z > 0

ð11Þ

for the anisotropic substrate:

j2z e2

xzs � exxsezzs

� �þ e1se3sexxs < 0 and

e1s � j2z < 0

ð12Þ

and for the isotropic cover:

e1c � j2z < 0: ð13Þ

Using (11) and (12) and the definitions in (2), pure homo-geneous hybrid waves with real jz can be captured, in terms ofh and /, by the following inequalities.

Figure 4. The change in the regions defining pure hybrid homogeneous waves for the anisotropic guide with all-positive diagonal tensorelements as the angles of orientation are changed. Values used: e1s = 3.3856, e3s = 2.25, e1f = 4 and e3f = 2.7889. The blue areas show howchanging the value of / from / = 0� (the blue curves) to / = 90� (the red curves) has little effect on the range of h over which pure hybridhomogeneous modes are possible.

4 A.D. Boardman et al.: EPJ Appl. Metamat. 2015, 2, 11

For the extraordinary mode:

e3se1s þ e3s � e1sð Þcos2/sin2he3s � e3s � e1sð Þsin2hsin2/

< j2z

< e3fe1f þ e3f � e1f

� �cos2/sin2h

e3f � e3f � e1f

� �sin2hsin2/

ð14Þ

and for the ordinary mode:

e1c < j2z < e1f ð15Þ

For a substrate with parameters e1s = 3.3856, e3s = 2.25,and a thin film guide with parameters e1f = 4, e3f = 2.7889,the effect of changing the value of / is very small as can beseen in Figure 4. Equations (11) and (12) give the blue curveswhen / = 0�, and the red curves when / = 90�. Applying equa-tions (14) and (15) produces the areas in blue from which it canbe seen that the change in / has little effect on the range of hover which pure homogeneous modes occur.

Now the inequalities given in (14) and (15), permitting theexistence of various types of waves, are quite general. For a

Figure 5. The figure demonstrates the increasing range in h over which homogeneous waves occur in a hyperbolic film with e3f < 0, as /increases towards a certain critical value. The dark blue areas show the regions over which pure hybrid homogeneous guided wave arepossible.

Figure 6. The graphs show the range of h over which homogeneouswaves can propagate for e1s = 3.3856, e3s = 2.25, e1f = 4,|e3f| = 2.7889. The range extends from h = 0� up to each of thecurves in the figure. As / increases, there is little change whene3f > 0. When e3f < 0, there is a rapid increase in the range until, at aparticular value / ¼ /crit, homogeneous waves can propagate at allangles of h up to 90�. With the parameters given in the text/crit ¼ 39:9� (shown by the dotted line). The upper right hand curveshows the decreasing range of h when u increases beyond / ¼ /crit.

Figure 7. The dependence of the critical angle (/c in the figure) onthe value of e3f < 0, above which the wave number jz is no longerlimited by the value of e3f when this is negative. When e3f > 0 thevalue of j2

z must be less than e3f for propagation of hybridhomogeneous modes. If e3f < 0, there will be an angle /crit abovewhich there will be propagation of inhomogeneous modes.The arrow in the diagram shows the critical angle whene3f = �2.7889.

A.D. Boardman et al.: EPJ Appl. Metamat. 2015, 2, 11 5

hyperbolic film, where e3f < 0, the inequality on the right handside of (14) becomes

j2z < e3f

�� �� e1f � e3f

�� ��þ e1f

� �cos2/sin2h

e3f

�� ��� e3f

�� ��þ e1f

� �sin2hsin2/

: ð16Þ

The profound effect of this change can be demonstrated, indetail, by looking analytically at the changes in the regionswhere the hybrid homogeneous, inhomogeneous and leakymodes can exist. In order to illustrate the specific effect of anegative value for e3f , the value e3f ¼ �2:7889 will beselected, and the other permittivity values will remain thesame. The effect of this change in sign is shown in Figure 5,and contrast greatly from the results shown in Figure 4.

From Figure 5, it is apparent that the pure hybrid homoge-neous mode region depends on the point where the curvesgiven by

j2z ¼ e3f

�� �� e1f � e3f

�� ��þ e1f

� �cos2/sin2h

e3f

�� ��� e3f

�� ��þ e1f

� �sin2hsin2/

ð16aÞ

for the different values of / cut the line j2z ¼ e1s. These

points are given when h = h/, where, for e3f > 0,

h/ ¼ sin�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie3f e1f � e1s

� �e1f � e3f

� �e3f cos2/f þ e1ssin2/f

� �s

ð17Þ

which, as is apparent in Figure 4, decreases, but not very sig-nificantly, as / changes from 0� to 90�. For the hyperboliccase, for which e3f < 0, (17) can be written

h/ ¼ sin�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie3f

�� �� e1f � e1s

� �e3f

�� ��þ e1f

� �e3f

�� ��cos2/f � e1ssin2/f

� �s

: ð18Þ

There is a dramatic difference in the two cases. When/ = 0�, the ranges of h over which pure hybrid homogeneouswaves propagate extend to

hmax ¼ sin�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie1f � e1s

e1f � e3f

rðfor the case where e3f > 0Þ ð19Þ

and

hmax ¼ sin�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie1f � e1s

e3f

�� ��þ e1f

sðfor the case where e3f < 0Þ ð20Þ

Comparing (19) and (20) it is clear that when e3f > 0� and/ = 0�, so that the optic axis is in the xz-plane, the range forthe anisotropic guide is wider than that for the hyperbolicguide. In the former case, as / increases from zero, it is easyto see from (18) that for e3f > 0, as e3f

�� ��cos2/f ! e1ssin2/f ,the range of h over which pure hybrid homogeneous modes canexist decreases, but only by a relatively small amount (Figures4 and 6). In contrast to this case, when e3f < 0 and / increasesfrom zero, the range increases rapidly until, as / approaches acertain critical value /crit, given by

/crit ¼ sin�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie3f

�� ��e3f

�� ��þ e1f

s; ð21Þ

pure hybrid homogeneous modes can exist for all values ofthe angle h from 0� to 90� (Fgures 5 and 6). Note that as/ approaches the value /crit, the change in the value ofhmax becomes very rapid so that very small changes in theorientation of the optic axis has a profound effect, whichcould have important consequences for the design and man-ufacture of hyperbolic guides.

It is apparent from Figure 6 that for the case where e3f > 0,the critical value of / has no meaning. It should be noted herethat the critical value also depends crucially on the value of e3f

as shown in Figure 7.A second important major difference between the usual

anisotropic case and the hyperbolic case is that in the lattermodes can propagate at values of j2

z > e1f . However, in thisregion the condition for the ordinary wave in equation (2),

Figure 8. Above critical angle /c, the normalised wave number jz is no longer limited by the value of e1f, but, in the regions above e1f andbelow the solid curves, the modes are inhomogeneous. The blue areas are the regions where hybrid homogeneous modes occur. The figureshows the decreasing range of h over which these modes are possible as / increases beyond the value of /c. The dashed line is the result forthe anisotropic substrate where as / changes the change in the graph is undetectable on the scale of the figure.

6 A.D. Boardman et al.: EPJ Appl. Metamat. 2015, 2, 11

namely, e1f � j2z > 0 is no longer satisfied, and the waves are

inhomogeneous. These waves and the associated leaky waveswill be examined in detail in a future paper.

As the angle / increases beyond the value of /c, the rangeof h over which pure hybrid homogeneous waves propagatedecreases again and is given by:

0 < h < sin�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie3f

�� ��e3f

�� ��þ e1f

� �sin2/

sð22Þ

Figure 8 demonstrates this decreasing range of range of hover which these modes propagate as / increases from a valuejust above /c to the angle / = 90�.

The above analysis shows, for a particular data set, howvarying the position of the optic axis leads to interesting con-clusions about the regions of existence of different forms ofwaves. In particular it shows how varying the orientation ofthe optic axis creates, or closes, windows of opportunity topropagate the kind of waves that are certain to be part ofupcoming devices. In order for a meta-device to operate in ahomogeneous wave window it is important first of all to con-duct an investigation into how the orientation angle / can bemade to influence the wave outcomes. The analysis also intro-duces the existence of inhomogeneous waves for values of thenormalised wave-number that lie above the maximum possibil-ities for the anisotropic guide with a permittivity tensor with allpositive elements.

4 Conclusions

As opposed to what is revealed in the limited literature onoptic axis orientations in metamaterials, the general theorygiven here shows the advantages of orienting the optic axisin carefully defined directions.

Through examining the dispersion relations for hyperbolicwaveguides, this paper emphasises that the orientation of theoptic axes is a significant determinant in the resulting opticalproperties. In fact, whereas for conventional anisotropic wave-guides, homogeneous propagating waves occur over a limitedrange of the polar angle h, a range that varies only a little withthe azimuthal angle /, we have shown that for a hyperbolicguide, a critical angle /c exists, at which value the guidedwaves are homogeneous for all values of h between 0� and90�. This has significant implications for meta-waveguidedesigns in that it implies that, in the hyperbolic case, genera-tion of propagating modes can be more easily achieved, andthe resulting structures are more tolerant to optic axismisalignment.

The high sensitivity of the mode propagation constant toazimuthal orientation of the optic axis could also be

transformative for meta-guided wave technology. Small exter-nal perturbations, such as the application of mechanical pres-sure, or temperature, can result in changes to the effectiveorientation of the guide/substrate axes. These in turn, willresult in modes being cut off, or allowed to propagate, openingup potential for sensor applications. Indeed, electro-optic con-trol over the orientation of the optic axis in a hyperbolic guidecould lead the way to a communications technology based onactive mode control.

Also, we note the implication of our work for transforma-tion designs that exploit the orientation of the optic axis ofanisotropic media, as described in [13]. In that work, pixelsof anisotropic media, where the orientation of the optic axisalternated between pixels, were shown to produce an effectivetransmission medium capable of a variety of transformationdevices, including a wave expander and a source shifter. Ourwork suggests that a greater range of applications is possiblethrough the use of pixelated hyperbolic media.

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Cite this article as: Boardman AD, Egan P & McCall M: Optic axis-driven new horizons for hyperbolic metamaterials. EPJ Appl. Metamat.2015, 2, 11.

A.D. Boardman et al.: EPJ Appl. Metamat. 2015, 2, 11 7