. INTRODUCTIONecently, media that exhibit a negative index of refractionave been the topic of extensive work.1–4 This is due tohe development of the first negative index medium bymith et al.5 and the subsequent explosion of publicationsn the subject that followed. An important characteristicf negative index media is that the wave vector k, thelectric field vector E, and the magnetic field vector Horm a left-handed triplet, a relationship that has ledhem to be dubbed left-handed media (LHM). The afore-entioned relation among the three vectors implies that
he Poynting vector �S� and the wave vector �k� of a waveropagating in a LHM are necessarily antiparallel, whichs the defining characteristic of the so-called backwardave. Therefore, LHM and consequently media withegative permittivity and permeability (negative index)re in essence backward wave media; but what about theeverse? Does an antiparallel relationship between theoynting vector and the wave vector imply a unique nega-ive index of refraction? And if so, then what would be theunctional form of this index?
In this paper, the backward-wave phenomenon will betudied from a purely three-dimensional wave propaga-ion point of view. To do so, it is convenient to restrict ourttention to propagation through a transmission pass-and (i.e., away from regions of anomalous dispersion). Inuch a passband, the group velocity and Poynting vectoroint in the same direction and the problem can be for-ulated in terms of the group velocity instead of theoynting vector. Furthermore, it is also convenient to sub-titute the propagation vector with the phase velocity. Ithould be noted that these substitutions are appropriaten the context of the present work, since we are only con-
erned with the directions of the vectors and not necessar-ly their magnitudes.
Our analysis of backward-wave propagation and its re-ation to the index of refraction will be presented as fol-ows. In Section 2 the index of refraction necessary to pro-uce backward wave behavior is derived using theondition that the group and phase velocities are perfectlyntiparallel (the angle between them is 180°). It is shownhat this index is necessarily negative and isotropic. Theondition that the phase and group-velocity vectors areerfectly antiparallel is then relaxed and the case is con-idered where the two vectors have at least one antipar-llel component. In this case the medium is necessarilynisotropic requiring a negative index along at least onerincipal axis. Moreover, constraints on the functionalorm of the index of refraction are obtained using the con-ition that the group velocity must be positive in theransmission passband. Anisotropic uniaxial right-anded media (RHM) and LHM are further examined inection 3 where the angles between the phase and groupelocities under certain conditions are calculated. Sectioncontains our final thoughts and conclusions.
. BACKWARD WAVES. Perfect Backward Waveso examine the backward-wave phenomenon we startith a purely three dimensional wave propagation pointf view, considering an arbitrary medium with phase in-ex n�k�. The challenge is to find a functional form for thehase index that results in the propagating wave exhibit-ng backward behavior. To find this functional form, weegin with a known property of backward waves. That is,
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2378 J. Opt. Soc. Am. B/Vol. 23, No. 11 /November 2006 J. Woodley and M. Mojahedi
he phase and group velocities of such waves must be an-iparallel, where the phase and group velocities are giveny
vp =c0
n�k�k̂, �1�
vg = �k
c0k
n�k�. �2�
ere, co is the speed of light in vacuum and k̂ is the unitector in the direction of propagation. In an arbitrary me-ium where the relationship between the group velocitynd phase velocity is not known, the two vectors can beelated by
�vgx,vgy,vgz� = �dxvpx,dyvpy,dzvpz�, �3�
here the di �i=x ,y ,z� are arbitrary constants that ac-ount for the differences in magnitude and direction be-ween the components of the phase and group velocities.ote that if any of the components of the phase velocityre zero, Eq. (3) may not be valid. However, a new coordi-ate system can always be defined in which the phase ve-
ocity has no zero components. For the case of a perfectackward wave medium, the angle between the phasend group velocities must be 180°. This condition has twoonsequences for Eq. (3). First, we must choose dx=dydz=d so that the vectors are collinear. Second, d must beegative to ensure that the vectors are perfectly antipar-llel. To enforce the fact that d must be negative, it will beritten in the form −�d�. Hence, Eq. (3) becomes
vg = − �d�vp. �4�
sing Eqs. (1) and (2) for the group and phase velocitiesn Eq. (4) gives
�k
c0k
n�k�= − �d�
c0
n�k�k̂. �5�
he parameter d, as written in Eq. (5), is now an arbi-rary constant that accounts for any difference in theagnitudes of the two velocity vectors. It is interesting to
ote that the phase index n�k� that satisfies Eq. (5) willesult in backward wave propagation for all frequencies.owever, it should be noted that this result is purelyathematical in the sense that no physical medium cur-
ently exists that satisfies Eq. (5) for all frequencies. Set-ing d=1 for simplicity, the solution to the above vectorifferential equation is given by
n�k� = bk2, �6�
here b is an arbitrary constant with units of squareeters. The dispersion relation corresponding to this in-
ex is
��k� =c0k
n�k�=
c0
bk. �7�
igure 1 shows the dispersion diagram calculated fromq. (7). The wave vector k in branch I of Fig. 1 is negative,
mplying a negative phase velocity. On the other hand,he local derivative at any point on branch I is positive,
ignifying that the group velocity is positive everywhere.n branch II the opposite is true. That is, at all points thehase velocity is positive and the group velocity is nega-ive. Hence, as expected, the index derived from Eq. (4)ields backward wave behavior at all frequencies sincehe phase and group velocities are antiparallel every-here.It is important at this point to determine which branch
n Fig. 1 presents a valid solution. Assume that the propa-ating waves are generated by a fixed source located at=0 and that the energy of the waves propagates away
rom the generator in the +r̂ direction. For this configura-ion, the solution represented by branch I implies a groupelocity that propagates away from the source (+r̂ direc-ion) and a phase velocity that propagates toward theource (−r̂ direction). The solution represented by branchI, on the other hand, implies the opposite case with theroup velocity propagating toward the source (−r̂ direc-ion) and the phase velocity propagating away from theource (+r̂ direction). Since we have assumed propagationn the passband, where the group and energy velocitiesre the same, the solution in branch II directly violatesur assumption that the energy of the wave propagatesway from the source. Therefore, the situation in branchI is unphysical and does not present a valid solution.his can also be seen as follows. Branch II represents thease of a negative group velocity. It has previously beenhown that in passive media the group velocity can beegative only in the stop band.2,6,7 Hence, branch II isgain unphysical since we are assuming propagationhrough the passband of a passive medium. It should beoted that branch I can be obtained only by choosing b0 in Eq. (7). However, choosing b�0 is also the condi-
ion used to obtain a negative index of refraction in Eq.6). Therefore, we must conclude that for the case consid-red above, a negative index of refraction is a necessaryondition for backward wave propagation. We may takeote of the fact that the dispersion relation obtainedbove is in agreement with those derived for the dual of aimple transmission line model.8 This model, which con-ists of a series capacitor and a shunt inductor (instead ofcustomary series inductor and shunt capacitor), was the
tarting point for several of the first negative index mediaeveloped to date.9,10 To our knowledge, this is the firstttempt to derive these results from the full wave propa-ation (3D) point of view.
ig. 1. (Color online) Dispersion relation for the index given inq. (7).
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J. Woodley and M. Mojahedi Vol. 23, No. 11 /November 2006 /J. Opt. Soc. Am. B 2379
. Imperfect Backward Wavesn Subsection 2.A we considered a case where the phasend group-velocity vectors were perfectly antiparallel (thengle between them was 180°). To perform a more generalnalysis, the angle between the phase and group veloci-ies should be allowed to have any value in the range of0° ���270°. In other words, we should allow the dotroduct of the unit vectors of the phase and group veloci-ies to have any negative value. We will refer to these asmperfect backward waves. For such a wave, although thehase and group velocities are not antiparallel, they willave antiparallel components. Hence, the point of refer-nce is once again Eq. (3), but now the di’s can no longerll be identical. In terms of the components of the groupnd phase velocities, from Eq. (3) we can write
vgi = divpi, �8�
here i=x ,y ,z. If we use the expressions of Eqs. (1) and2) for the group velocity and phase velocity, Eq. (8) can beritten as
c0� 1
n�k�−
k
n�k�2
�n�k�
�ki�ki
k= di
c0
n�k�
ki
k. �9�
ith a little algebra Eq. (9) can be rewritten in the form
�1 − di�
n�k�=
k
n�k�2
�n�k�
�ki. �10�
he solution to the above differential equation is given by
n�k� = �i�k2��1−di�/2, �11�
here �i is an arbitrary constant. Equation (11) showshat for this imperfect backward wave the index can haveifferent forms along each of the three principal axes,hich is the case for an anisotropic medium. This will be
urther examined in Section 3 where it will be shownhat, to obtain imperfect backward wave propagation, thendex along at least one of the axes must be negative.hat is, we must have �i�0 for at least one of the solu-
ions in Eq. (11). Note that the simple case of an isotropicedium can be recovered from Eq. (11) by setting di=d−1.
. Minimum Required Dispersion in the Case ofegative Index Metamaterials
n the cases presented above a phase index was obtainedhat yielded backward wave behavior at all frequencies.urther insight into the possible functional forms andonstraints on the phase index can be gained by rewritinghe group velocity according to
vg = � c0
n�k�−
c0k
n�k�2
�n�k�
�k �k̂. �12�
ince, as discussed previously, group velocity must beositive in the passband, Eq. (12) provides us with a con-traint on the functional form of the index. In otherords, any solution for the index that yields propagatingackward waves must also give a positive result for theroup velocity when substituted in Eq. (12). As an ex-mple let us assume the index can be written in the form
n�k� = �kp, �13�
here � and p are arbitrary constants. Substituting Eq.13) into Eq. (12), the conditions for a positive group ve-ocity become
p � 1 for � � 0, �14a�
p � 1 for � � 0. �14b�
ote that the phase index in Eq. (6) has the form given byq. (13), which satisfies the condition of inequality (14a),ince p=2 and �=b�0. In other words, substituting Eq.6) into Eq. (12) yields
vg = −c0
bk2 � 0. �15�
he index given by Eq. (6) is isotropic. In the anisotropicase the constraints of inequalities (14a) and (14b) muste applied separately to each direction in which there is aassband. Using Eq. (11) in conjunction with inequalities14), we now have a set of conditions that provide theramework for generating an index of refraction, eithersotropic or anisotropic, which will produce the backwardave phenomenon. Finally, the conditions given in Eq.
13) and inequalities (14) also demonstrate an importantharacteristic of negative index media: In the negative in-ex case ���0�, the medium is necessarily dispersive �p1�.
. ANISOTROPIC MEDIAn Section 2 it was shown that a necessary condition for aerfect backward wave is an isotropic index of refraction.urthermore, an analysis of the dispersion characteristicsetermined that the index was necessarily negative. Inhe case of the imperfect backward wave it was shownhat the index could not be identical along all three prin-ipal axes. In other words, the medium must be aniso-ropic. However, no constraints on the sign of the indexere established. In this section a general analysis will beerformed to determine the requirements on the sign ofhe index along the principal axes. For simplicity let usonsider a uniaxial medium where the permittivity andermeability are given by
�� = �0��s 0 0
0 �s 0
0 0 �z , �16�
� = �0�r, �17�
here �0 and �0 are the free-space permittivity and per-eability, respectively. From Eqs. (16) and (17) the optical
xis of the uniaxial medium is directed along the z axis.n this case, a so-called ordinary wave will correspond tony wave propagating such that its D vector is perpen-icular to the optical axis (i.e., D polarized in the x–ylane). In uniaxial media, ordinary wave propagation hasn isotropic character. Since this type of behavior was ex-mined in Section 2, the ordinary wave will not be consid-red here. If, on the other hand, the D vector is polarized
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2380 J. Opt. Soc. Am. B/Vol. 23, No. 11 /November 2006 J. Woodley and M. Mojahedi
n the plane containing both the optical axis and theropagation direction, then the wave is considered ex-raordinary. The k-space diagram for the extraordinaryave propagating through a medium with material pa-
ameters �s=1, �z=−2, and �r=1 is shown in Fig. 2. Ithould be noted that since in the following analysis were mainly concerned with the shape of the k surfaces,nd since different values of � only scale these surfaces, aonvenient value of � can be chosen. For simplicity, thelots in this section are generated using �=1.For an extraordinary wave propagating along one of
he three primary axes, Fig. 2 can be used to further elu-idate the situation. For propagation in the x direction thextraordinary wave is polarized along the z axis. For thisolarization, the index is imaginary so that there is noropagation in the x direction. Similarly, the index for anxtraordinary wave propagating in the y direction willlso be imaginary, cutting off propagation. On the otherand, an extraordinary wave with a propagation vectorlong the z axis (along the optical axis) will be polarizedn the x–y plane, where the index is real and positiveRHM), and it will propagate. This is the point at whichhe k surfaces for the ordinary and extraordinary waves
ig. 2. (Color online) (a) 3D and (b) two-dimensional (2D)-space diagrams for RHM and LHM two-sheeted hyperboloids.n both cases the media are uniaxial with �x=�y=�s. In the RHMase the parameters are �s=1, �z=−2, and �r=1. The parametersn the LHM case are �s=−1, �z=2, and �r=−1. Because the pa-ameters in the LHM case are simply the negatives of those inhe RHM case, the k surfaces are identical. The angle betweenhe phase velocity and group velocity for the RHM and LHMases are shown on the 2D plots.
ntersect so that the two types of waves cannot be distin-uished. If we now negate all the parameters used to gen-rate the k surface (i.e., �s=−1, �z=2, and �r=−1), the re-ulting k-space diagram will be identical to that in Fig. 2.owever, in this case waves propagating in the z directionill see a negative index (LHM) instead of a positive onehereas waves propagating in the x or y directions willgain be cut off. The k surface in Fig. 2 is an example of awo-sheeted hyperboloid that results when the extraordi-ary wave in a uniaxial medium only propagates in theirection of the optical axis.Figure 3 shows an example of a one-sheeted hyperbo-
oid. This type of k surface results in a medium wherenly a wave with D polarized in the direction of the opti-al axis can propagate. This implies that there is noropagating ordinary wave in this medium since, by defi-ition, the D vector for the ordinary wave must be polar-
zed perpendicular to the optical axis. The k surface ofig. 3 corresponds to an extraordinary wave for a mediumith parameters �s=−1, �z=2, and �r=1 in the RHM case,r �s=1, �z=−2, and �r=−1 in the LHM case. In bothases, for propagation in the x and y directions, the ex-
ig. 3. (Color online) (a) 3D and (b) two-dimensional (2D)-space diagrams for RHM and LHM one-sheeted hyperboloids.n both cases the media are uniaxial with �x=�y=�s. In the RHMase the parameters are �s=−1, �z=2, and �r=1. The parametersn the LHM case are �s=1, �z=−2, and �r=−1. Because the pa-ameters in the LHM case are simply the negatives of those inhe RHM case, the k surfaces are identical. The angle betweenhe phase velocity and group velocity for the RHM and LHMases are shown on the 2D plots.
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J. Woodley and M. Mojahedi Vol. 23, No. 11 /November 2006 /J. Opt. Soc. Am. B 2381
raordinary wave is polarized along z where it sees a posi-ive index in the RHM case or a negative index in theHM case. For z-directed waves the extraordinary wave
s polarized in the x–y plane. These waves are attenuatedue to the fact that the permittivity and permeabilityave opposite signs. Table 1 summarizes the propagationehavior for the one-and two-sheeted hyberboloids.As shown above, the k surface shapes do not change
hen the signs of the parameters �s, �z, and �r are re-ersed. In other words, the RHM surfaces are indistin-uishable from the LHM surfaces for both the one- andhe two-sheeted hyperboloids considered. Hence, if we areaced with the task of determining the sign of the index ofefraction, the information provided by a visual inspec-ion of these equifrequency surfaces alone is insufficient.o determine the sign of the index we must return to annalysis similar to that performed in Section 2, that is, weust examine the relationship between the phase and
roup-velocity vectors.For both k surfaces a cut was taken in the ky–kz plane
nd the angle between the phase and group-velocity vec-ors was calculated as a function of kz. Figure 4(a) showshe angles resulting from the two-sheeted surface of Fig., and the angles calculated for the one-sheeted surface ofig. 3 are shown in Fig. 4(b). For the RHM cases (solidurves) of both the one- and two-sheeted hyperboloids thengle between the phase velocity and group velocity be-ins at 0° at the onset of propagation (kz=1 for the two-heeted hyperboloid and kz=0 for the one-sheeted hyper-oloid) and approaches 90° asymptotical from below as kzecomes large. On the other hand, in the LHM casesdashed curves) the angle begins at 180° at the onset ofropagation and approaches 90° asymptotical from aboven the large kz limit. Therefore, for both uniaxial mediaonsidered, the dot product between the unit vectors ofhe group velocity and phase velocity is always positive inhe RHM cases and negative in the LHM cases. Hence,he LHM versions of the one- and two-sheeted hyperbo-oids both exhibit imperfect backward wave propagation.n addition, an analysis of the components of the phasend group velocities in the LHM cases for both structureshows that their ky components are directed in the sameirection and their backward wave behavior is therefore aesult of oppositely directed kz components. This empha-izes that imperfect backward wave behavior does not re-uire all of the components of the phase and group veloci-ies to be antiparallel.
In the above analysis the relationship between thehase velocity and group velocity in anisotropic LHM andHM was discussed using equifrequency surfaces. Inther words, the permittivity and permeability valuesere taken at a fixed frequency and were constant. A
Table 1. Index Seen by Different Polarizations inthe Two- and One-Sheeted Hyperboloids
k Surface Two-Sheeted One-Sheeted
olarization x–y plane z(optical axis)
x–y plane z(optical axis)
Index ±1 j2 j ±2
uestion can then be asked in regard to the influence ofemporal dispersion on the above analysis. To answer thise must keep the following in mind. First, the geometryf the k surface (an ellipsoid, a one-sheeted hyperboloid, awo-sheeted hyperboloid, or a null surface for which noropagation is allowed) is determined by the signs of thearameters � and �. Second, the dimensions of the k sur-aces (i.e., the vertices of the ellipsoid or hyperboloid) areetermined by the magnitudes of � and �. Therefore, ifhe parameters � and � are changing with frequency, in-tead of a single k surface there will be a family of k sur-aces, each taken at a different frequency. In the fre-uency regions where the parameters do not changeigns, each of these k surfaces will have the same geom-try (i.e., ellipsoid, hyperboloid, no propagation) so thathe sign of the dot product between the group velocity andhase velocity, and hence the general relationship be-ween them (left or right handed), will not change sign.
hen one of the parameters changes signs, the geometryhanges (for example, from a one-sheeted hyperboloid to awo-sheeted hyperboloid) and then a new relationship be-ween the group and phase velocities emerges. This rela-ionship is maintained until one of the parametershanges signs once again. Therefore, for the purposes ofur analysis here, it is sufficient to analyze the general
ig. 4. (Color online) Calculated angle between the phase androup-velocity vectors for the RHM and LHM hyperboloidal kurfaces considered. (a) Two-sheeted hyperboloid (Fig. 2). (b)ne-sheeted hyperboloid (Fig. 3).
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2382 J. Opt. Soc. Am. B/Vol. 23, No. 11 /November 2006 J. Woodley and M. Mojahedi
ehavior of the four possible geometries (a single ellip-oid, a one-sheeted hyperboloid, a two-sheeted hyperbo-oid, and a null surface) in the absence of dispersion, sincehe effects of the dispersion are only important to the ex-ent that they can result in a change in the signs of theermittivity and permeability. The general case of an el-ipsoid was considered in Section 2 and the one- and two-heeted hyperboloids were examined in this section.
. CONCLUSIONhe problem of backward wave propagation was studied
rom a purely three-dimensional propagation point ofiew and two different types of backward waves were de-ned: perfect and imperfect. In a perfect backward wavehe phase and group-velocity vectors are antiparallel (i.e.,he angle between the vectors is 180°). In the second type,he imperfect backward wave, the dot product betweenhe phase and group-velocity vectors is negative. Startingith the above relationships between the phase androup-velocity vectors, the forms of the indices necessaryo produce these types of backward waves were obtained.n both cases, it was shown that backward wave behaviorould only be the result of propagation through a negativendex medium. In the case of the perfect backward wave,he index was necessarily negative and isotropic while forhe imperfect backward wave the medium was aniso-ropic possessing a negative index along at least one ofhe three principal axes. Constraints on the functionalorm of the indices were then obtained using the conditionhat the group velocity must be positive in a transmissionassband. Although at the moment the construction ofeft-handed media at optical frequencies remains a diffi-ult task, such media could find use as beam-steering de-
ices and perfect optical lenses. In addition, for the aniso-ropic case, the polarization dependence of the mediaould be taken advantage of to make filters, polarization-ependent lenses, and beam-steering components.
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