PlasmonicsDOI 10.1007/s11468-014-9739-7

Optical Propagation Through Graded-Index Metamaterialsin the Presence of Gain

Ivan D. Rukhlenko

Received: 24 March 2014 / Accepted: 20 May 2014© Springer Science+Business Media New York 2014

Abstract We present a comparative theoretical study ofelectromagnetic wave propagation through randomly struc-tured isotropic and stratified anisotropic active metamaterialslabs, whose refractive indices vary smoothly from positivevalues on one end of the slabs to negative values on theother. The chief advantage of our two configurations overrecent hypothetical structures is that they can be made fromreal materials using common fabrication techniques. Wefind that even a small excess of gain (loss) in the randomlystructured slab can result in a strong resonant amplification(absorption) of incident field on the length scale of an opti-cal wavelength. This collective effect—somewhat similar toLandau amplification (damping) in a collisionless plasma—may prove useful in making compact optical amplifiers andperfect light absorbers.

Keywords Metamaterials · Electromagnetic optics ·Subwavelength structures · Effective medium theory ·All-optical devices

Introduction

Over the past few years, the emerging field of pho-tonic metamaterials has grown rapidly through continuingadvances in the fabrication of artificial composite struc-tures [1–5]. Metamaterials are of significant fundamental

I. D. Rukhlenko (�)Monash University, Melbourne, Victoria, 3800, Australiae-mail: ivan.rukhlenko@monash.edu

I. D. RukhlenkoITMO University, Saint Petersburg 197101, Russia

and practical interest owing to their unusual electromag-netic response, e.g., the possibility of a negative refrac-tive index or giant optical activity with negligible circulardichroism [6–8]. Moreover, their response can be engi-neered in practice by altering the design of the so-calledmeta-atoms—subwavelength building blocks of metama-terials [9–11]. It is believed that photonic metamaterialswill affect many aspects of our lives in the not-too-distantfuture through their novel applications in slow-light optics,sensing, imaging, and defence [12–14]. A variety of inte-grated photonics devices can benefit from graded-indexheterostructures in which the refractive index varies spa-tially in one or more directions [15–17]. In this work,we study transmission through two fundamental types ofsuch heterostructures fabricated using only homogeneousand isotropic materials, with an emphasis on the impact oftopology and gain on their optical response to the incidentlight.

Graded-Index Metamaterials

Consider a metamaterial slab of thickness h, whose permit-tivity and permeability smoothly vary from εa and μa atthe interface z = 0, to εb and μb at the interface z = h.Such a slab can be fabricated to have different patternsusing two materials (a and b) with constant parameters (εa ,μa for material a and εb, μb for material b) [14]. Specif-ically, a stratified pattern is obtained when one depositsalternative layers of two materials, gradually decreasing thethickness of layer a from h0 to zero while simultaneouslyincreasing the thickness of layer b from zero to h0. Such aheterostructure is schematically shown in Fig. 1a. The rel-ative volume fraction of material b is characterized in thez direction by filling factor f (z) such that f (0) = 0 and

Plasmonics

f (h) = 1; f gives the volume fraction occupied by mate-rial b within a thin layer h0 � �z � h between planesz and z + �z. If h0 is much smaller than the wavelengthof light, λ, and the function f (z) does not change signifi-cantly on the length scale of h0, i.e., h0 � max(λ, 1/|f ′|),then the effective permittivity and permeability of the strati-fied slab can be described with the effective-medium theory[8]. According to this theory, the entire slab behaves as auniaxial anisotropic material characterized by the diagonalpermittivity and permeability tensors with components

η‖(z) = ηa[1 − f (z)] + ηbf (z), (1a)

η⊥(z) = ηaηb

ηaf (z)+ ηb[1 − f (z)] , (1b)

where η = {ε, μ} and the subscripts ‖ and ⊥ designatedirections parallel and perpendicular to the slab’s plane.

Alternatively, an isotropic metamaterial slab shown inFig. 1b can be created in a randomly structured compositeof materials a and b with a gradually changing filling factor.Such a composite is realized by varying the relative concen-tration of the two materials during the fabrication process.If the typical size h0 of the composite’s inhomogeneity ismuch smaller than λ and the filling factor satisfies the con-dition h0|f ′| � 1, then the permittivity and permeability ofthe slab are given by the Bruggeman’s formula [14, 18]

η(z) = 1

4

{(2−3f )ηa + (3f −1)ηb ±

√[(2−3f )ηa + (3f −1)ηb

]2 +8ηaηb

}, (2)

in which the plus or minus sign is chosen to ensure the con-tinuity of η(z) within the slab while satisfying the boundaryconditions η(0) = ηa and η(h) = ηb. Although it is alwayspossible to separate the required analytical branch of func-tion η(z) for any given values of ηa and ηb , the expressionthat solves this problem for the general case of arbitrarycomplex variables ηa and ηb is still lacking [19].

It is well known that for certain designs of meta-atomsand excitation conditions, the effects of optical nonlocalitycan substantially alter the dispersion properties of metal–dielectric composites [20–24]. It should therefore be bornein mind that the following analysis will hold true only forthose graded-index slabs that may be prepared in such away that their optical properties are described by Eqs. (1)and (2) [25]. Also noteworthy is that the minimal scale ofthe slab’s inhomogeneity is determined by the size of theconstituting meta-atoms, which can be as small as a fewnanometers. The thickness h of the entire slab needs toexceed this scale by at least two orders of magnitude, toallow relatively smooth variations of the filling factor withinthe slab.

As a concrete example we consider the structure in whichmaterial a is a right-handed, positive-index material withηa = 1 + iδ, and material b is a left-handed, negative-index

material with ηb = −1+ iδ, where δ can be positive or neg-ative depending on whether the material is lossy or pumpedto provide some optical gain [13]. In Figs. 1c, d, we showhow ε(z) [or μ(z)] varies with z for the two types of slabsshown in Figs. 1a, b when the filling factor f (z) = z/h isincreasing linearly along the slabs’ thickness. The materialand geometric parameters are chosen to make the real partsof the functions in Eqs. 1a, b and 2 vanish at the center of theslabs. The anisotropic structure is characterized by η withtwo components: η‖ and η⊥. The dashed curves in panel cshow the imaginary part of η⊥ for δ = ±0.1 (the imaginarypart of η‖ is not shown). Solid curves show the real partsof the effective permittivity in both heterostructures. Here-inafter, we denote the real and imaginary parts of a complexnumber z using the notation z = z′ + iz′′.

Consider first the anisotropic case shown in Fig. 1c. Thequantity ε′‖ (solid blue curve) decreases linearly from +1 to−1, as one would expect for a graded-index metamaterial.At the same time, ε′⊥ (solid red curve) exhibits a shape thatis typical of an atomic resonance. Indeed, the imaginary partε′′⊥ (dashed red curve) shows a resonance-like structure thatrepresents loss or gain depending on the sign of δ. The gain(loss) is localized in the middle of the slab and its spatialdependence in the case of small δ is approximated by thefunction

ε′′⊥ ≈ δ

δ2 + (1 − 2f )2.

The area under the curve ε′′⊥(z) equals ±πh/2 in the limitδ → ±0, indicating that significant amplification of inci-dent light may occur in this limit. Note that the resonanceanalogy should be used with care here, because the loss orgain variations in Fig. 1c occur as a function of distance, notfrequency.

The optical response of the isotropic structure shownin Fig. 1d is somewhat different, because its ε and μ arescalars. The real part of ε (solid blue curve) again decreasesfrom +1 to −1, even though its variation is no longer per-fectly linear. The imaginary part ε′′ (dashed red curve) isnow bow-shaped, providing incident light with relativelyuniform gain or loss over the entire slab length dependingon the sign of δ. A somewhat surprising feature is that thiscurve does not change much with changes in the magnitudeof δ, and the area under the curve remains fixed. For theparameters of Fig. 1, the area under the curve ε′′(z) equals±πh/6 in the limit δ → ±0. This feature implies that sucha graded-index metamaterial slab may dramatically amplifythe transmitted beam even when its constituent materialsexhibits only a weak gain (small δ values). As we shallsee later, it also enables strong enhancement of the beamreflected from the slab. Hence, there is a significant differ-ence between the isotropic and anisotropic structures as faras their loss (or gain) distribution profiles are concerned.

Plasmonics

-1.0

-0.5

0.0

0.5

1.0

ε b, μ

b

-1.0

-0.5

0.0

0.5

1.0

-10

-5

0

5

10

Re(ε )

Re(ε )

δ = −0.1

εb, μ

bε2, μ

2εa, μ

aε1, μ

1

hh/2ε b

, μb

ε a, μ

a

ε2, μ

2ε1, μ

1

0 hh/20

Im(ε )

Im(ε ) Im(ε)(d) δ = −0.1

δ = +0.1Im(ε)

Re(ε)

(c)

δ = +0.1

zx

(b)(a) h0 h0 h0

Fig. 1 a Anisotropic and b isotropic graded-index slabs formed bytwo materials with constant permittivities (εa , εb) and permeabilities(μa , μb) and sandwiched between two homogeneous isotropic mate-rials with parameters ε1, μ1 and ε2, μ2. Panels c and d show spatial

variation of the real and imaginary parts of ε(z) [or μ(z)] when the fill-ing factor f (z) = z/h increases linearly along the slab of thickness h.In panel c, dashed curves show the imaginary part of ε⊥ for δ = ±0.1.The imaginary part of ε‖ is not shown for clarity

The strong amplification of an optical field by a graded-index isotropic slab composed of weakly active constituentmaterials is opposite in its nature to the well-known phe-nomenon of effective absorption by a macroscopic compos-ite made of lossless materials [25, 26]. In the latter case, theenergy absorbed by the composite is stored in local surfaceplasmon-polaritons (SPPs). In the event of amplificationstronger than that expected from the material gain alone,the energy transferred to the beam should come from exter-nal sources other than those pumping the active medium.If ε′′a = ε′′b = μ′′

a = μ′′b = δ, as was assumed in Fig. 1,

the enhancement of the mean square of the macroscopicelectric (or magnetic) field over the microscopic one ischaracterized by the factor [27]

(z) = η′′(z)δ

.

Note that η′′(z) also depends on the δ parameter.The important question is: what is the source of energy

for amplification stronger than what is expected from thematerial gain alone? We think that the situation is anal-ogous to that of Landau amplification in a collisionlessplasma [28], with the role of charge particles in plasmaplayed by the SPPs of metamaterials. In terms of this anal-ogy, amplification by the graded-index isotropic slab is acollective effect of energy exchange between the opticalwave and SPPs with an anisotropic momentum distribution.The energy exchange occurs only resonantly between SPPswith velocities v close to the phase velocity of the wavevph = (ω/k) ek , so that SPPs with |v| > |vph| increase the

energy of the wave and those with |v| < |vph| reduce it. Themechanism of SPPs excitation with an anisotropic momentdistribution depends on the specific metamaterial design.Since most metamaterials contain metals, it may be possibleto produce the required distribution by injecting a current orinducing the current by an external electromagnetic field. Itis clear that SPPs cannot freely propagate in the z directionof the anisotropic slab, which makes it impossible to satisfythe above resonance condition, except in the case of grazingincidence. That is why strong amplification of optical fieldoccurs in isotropic slabs and does not occur in anisotropicones.

Transmission Through a Graded-Index MetamaterialSlab

To further understand the amplification process, we con-sider the transmission of a transverse magnetic (TM) planewave, incident at an angle ϑ0, through the graded-indexmetamaterial slab shown in Fig. 1a. Propagation of thiswave in anisotropic medium is governed by the followingequation for the magnetic field:

∂2Hy

∂z2− 1

ε‖∂ε‖∂z

∂Hy

∂z+ ε‖

(μ‖k2 − β2

ε⊥

)Hy = 0, (3)

where β = n0k sinϑ0 is the transverse component of thewave vector, n0 is the refractive index of the surrounding

Plasmonics

medium, and k is the free-space wave number. The othertwo field components are given by

Ex = − i

ε‖k∂Hy

∂z, Ez = − β

ε⊥kHy. (4)

In the geometrical optics approximation, valid as longas the wavelength inside the slab is small comparedto the characteristic filling-factor variation length scale,Eq. 3 admits a simple analytical solution of the form[29]

Hy ∝ 1√Q

exp

(± i

∫ z

0Q(s) ds

), (5)

where Q = ±√ε‖

(μ‖k2 − β2/ε⊥

)and the branch of Q

is to be selected according to the handedness of the effec-tive medium at a particular point inside the slab [3] (i.e., thesign in front of the radical is generally different at differentpoints within the slab). If ε‖ = ε⊥ = ε and μ‖ = μ aregiven by Eq. 2, then the same equations describe an isotropicslab.

The geometrical optics approximation breaks down in thevicinity of the turning or “reflection” points at Q = 0 in theregion ReQ � |f ′|. In these regions,Q becomes imaginaryand the incoming wave is either attenuated or amplified,depending on the local optical activity of the slab. UsingEqs. 1a, b and 2, it is straightforward to show that the loca-tion of such points in the anisotropic slab is determined bythe relations

f = εa

εa − εb, f = εaμak

2 − β2

εa(μa − μb)k2 + (εa/εb − 1)β2.

(6)

In the isotropic slab, we obtain the relation

f = b ±√b2 + 4ac

2a, (7)

where a, b, and c are defined as

a = 3ξ2[2(εa − εb)ξ2 + (μa − μb)εaεb]

×[2(μa − μb)ξ2 + (εa − εb)μaμb], (8a)

b = (4ξ4 + εaεbμaμb)[(4εa − 3εb)μa − (3εa + 2εb)μb]ξ2 (8b)

+ 4ξ4[(2ε2a + ε2

b)μaμb + (2μ2a − 6μaμb + μ2

b)εaεb],

c = (1/3)(ξ2−εaμa)(4ξ2−εbμb)(2ξ

2+εaμb)(2ξ2+εbμa),

(8c)

and ξ = n0 sin ϑ0.

According to these relations, there are a maximum oftwo reflection points inside each slab if the filling factoris a monotonous function of z. The positions of the pointsmay vary with ϑ0, which affects the angular dispersion ofthe slab’s transmittance. Since permittivity and permeabil-ity are generally complex-valued whereas the filling factoris a positive real number not exceeding unity, for reflectionpoints to exist the four parameters εa , εb, μa , and μb mustobey certain conditions. For example, the first relation inEq. 6 holds only when sign(ε′aε′b) = −1 and ε′aε′′b = ε′′aε′b,whereas the second is satisfied for εbμb ≶ ξ2 ≶ εaμa .Unless the slab constituents are specifically selected to com-ply with these or other similar requirements, the reflectionpoints are absent. But even if such points exist, they arestill of little significance for anisotropic metamaterial com-posite, because parameter Q is generally complex in theirvicinity.

It should be recognized that the stratified anisotropic slabdoes not exhibit the effect of resonant field enhancement[15, 17] near the points where its dielectric permittivityand magnetic permeability both become zero (and in theisotropic slab such points are merely absent). Indeed, sup-pose that ε‖ = μ‖ = 0 at z = z0. In the vicinity of thispoint, we can write

ε‖ ≈ A(z− z0), μ‖ ≈ B(z− z0), ε⊥ ≈ εaεb

εa + εb,

and Eqs. 3 and 4 yield:

Hy ∼ cos[C(z − z0)

2], Ex ∼ (z− z0)2, Ez ∼ Hy,

where A, B , and C are constants. The transverse componentof the electric field of the TM wave is seen to vanish at pointz0, and the longitudinal component of the field is seen toremain finite even in the absence of absorption losses.

Using Eq. 5 and assuming that the metamaterial slabseparates two media with permittivities ε1, ε2 and perme-abilities μ1, μ2, we obtain the following transmission andreflection coefficients for the wave traveling from left toright (see Fig. 1):

T = 4(qa/qb)1/2 exp[i(ϕ − q2h)]ψ+a1ψ

+2b + ψ−

a1ψ−2b exp(2iϕ)

, R = ψ−a1ψ

+2b + ψ+

a1ψ−2b exp(2iϕ)

ψ+a1ψ

+2b + ψ−

a1ψ−2b exp(2iϕ)

, (9)

where qj = ±√εjμjk2 − β2, ϕ = ∫ h

0 Q(z) dz, and ψ±rs =

1±qr/qs . The coefficients of two linearly independent solu-tions in Eq. 5 are given by C± = (T /2)

√qbψ

±2b exp[i(q2h∓

ϕ)].It follows from Eq. 6 that the anisotropic slab contains

a reflection point f = 1/2 when it is formed by materialsof opposite handedness with εa = −εb and μa = −μb (ifone of these materials exhibits loss, the other must containgain). In this case, the exact solution of Eq. 3,

Hy ∼ exp

(± iqaz∓ 2iqa

∫ z

0f (s) ds

),

Plasmonics

Fig. 2 a Transmittance and breflectance spectra ofanisotropic (A) and isotropic (I)slabs made of absorptiveright-handed and amplifyingleft-handed materials, for fourvalues of dimensionlessparameter kh. Filling factorvaries linearly inside the slabs

is similar in form to Eq. 5 since factor Q−1/2 is a slowlyvarying function of z. As a result, coefficients T and R inthe present case are given by Eq. 9 with qa = −qb =√εaμak2 − β2, ϕ = qaheff, and the effective slab thickness

defined as

heff = h− 2∫ h

0f (z) dz. (10)

Results and Discussion

According to Eq. 9, in the absence of reflections from twointerfaces of the anisotropic slab (when ε1 = −ε2 = ε′a and

μ1 = −μ2 = μ′a), there are two possibilities for the slab

transmittance to reach unity: (i) when materials a and b aretransparent (i.e., ε′′a = μ′′

a = 0) or (ii) when loss is preciselycompensated by gain leading to heff = 0. In both cases,zero reflectance is possible because of constant impedance(μ‖/ε‖)1/2 along the slab and because function Q2 doesnot change sign at the reflection point f = 1/2. The latterfeature makes possible amplification of the transmitted andreflected waves inside an active isotropic slab when heff �= 0or ϕ �= 0.

For numerical calculations, we assume that gain in meta-material b slightly exceeds absorption in ordinary materiala, by choosing εa = μa = 1 + i0.01 and εb = μb = −1 −

Fig. 3 Transmittance (solidcurves) and reflectance (dashedcurves) of isotropicmetamaterial slab as a functionof its relative thickness, kh, forincidence angles of 0, π/6, andπ/3. Material parameters are thesame as in Fig. 2

Plasmonics

i0.02. We also choose ε1 = μ1 = 1 and ε2 = μ2 = −1 forthe two materials surrounding the slab to ensure that Fresnelreflection is absent at the two interfaces. Transmittance andreflectance are shown in Fig. 2a, b as functions of incidenceangle for four thicknesses (quantified by the dimension-less parameter kh) of anisotropic and isotropic metamaterialslabs. It is seen from the figures that a TM-polarizedwave is strongly amplified inside isotropic slabs for almostall incidence angles. In contrast to this, anisotropic slabsamplify only waves at grazing incidence, i.e. only wavestraveling long enough in the transverse direction insidethem.

If we modify the material parameters such that theloss in ordinary material a slightly exceeds the gain inmetamaterial b, the numerical results show almost com-plete absorption of the TM wave within the isotropic slab.But again, this configuration has little impact on the TMwave if it propagates through the slab that is anisotropic.It is important to note that the strong amplification (orabsorption) of light does not occur if materials a andb are of the same handedness, because in this case theimaginary parts of the effective permittivity and perme-ability remain small over the entire slab. Figure 2 alsosuggests that the TM wave can exhibit broadband ampli-fication in a graded-index structure as thin as the wave-length. For any particular angle of incidence, there is anoptimum thickness of the isotropic composite that cor-responds to the maximum transmittance, as indicated inFig. 3. For example, (kh)opt ≈ 9.5, 2.6, and 0.8 for ϑ0 =0, π/6, and π/3, respectively. The reflectance also peakswith increasing kh for oblique incidence, and saturatesas kh → ∞.

There are several challenges that need to be met before anisotropic metamaterial slab of the kind discussed here canprove beneficial for realistic integrated photonics devices.First of all, as was demonstrated in Section “Graded-IndexMetamaterials,” the actual values of reflectance and trans-mittance are determined not by the local material parametersof the slabs’ constituents, but rather by the process of macro-scopic gain saturation. That is why it is crucial to findefficient ways to excite SPP resonances inside the slabs,in order to amplify an incident wave with high gains seenin Figs. 2 and 3 (since the effect of gain saturation isneglected in our study of the continuous-wave propagation,the values of the reflectance and transmittance presentedin these figures should be considered the upper theoreticalbounds for what can be achieved with real optical pulsesin the presence of gain degradation). If such high gain lev-els are achieved in a tiny region of metamaterial slab, theylead to a considerable increase in the photon density ofstates and the resulting enhancement of the spontaneousemission. We leave theoretical treatment of such issues forfuture work.

Conclusions

We have theoretically examined optical propagation throughtwo graded-index metamaterial composites designed withdifferent topologies. Our analysis revealed that strongamplification of light can be realized inside isotropic struc-tures made of right-handed and left-handed materials, oneof which is active. The same structure fully absorbs light ifits constituent materials are passive. We also showed that,in contrast to previous reports [15, 17], the resonant fieldenhancement near point ε = 0 is absent in the transitionregion between the materials of opposite handedness in boththe isotropic and anisotropic composites considered in thispaper.

Acknowledgments This work was supported by the AustralianResearch Council through its Discovery Early Career ResearcherAward DE120100055.

References

1. Zhu W, Rukhlenko ID, Premaratne M (2012) J Opt Soc Am B29:2659

2. Zhu W, Rukhlenko ID, Premaratne M (2012) IEEE Photonics J4:741

3. Premaratne M, Agrawal GP (2011) Light propagation in gainmedia: optical amplifiers, Cambridge University Press, Cam-bridge

4. Veselago VG, Braginsky L, Shklover V, Hafner C (2006) JComput Theor Nanosc 3:1

5. Ramakrishna SA (2005) Rep Prog Phys 68:4496. Zhu W, Rukhlenko ID, Huang Y, Wen G, Premaratne M (2013) J

Opt 15:1251017. Rukhlenko ID, Premaratne M, Agrawal GP (2012) Adv Opto-

Electron 2012(907183)8. Rukhlenko ID, Premaratne M, Agrawal GP (2010) Opt Express

18:279169. Zhu W, Rukhlenko ID, Si L-M, Premaratne M (2013) Appl Phys

Lett 102:12191110. Zhu W, Rukhlenko ID, Premaratne M (2013) Appl Phys Lett

102:24191411. Zhu W, Huang Y, Rukhlenko ID, Wen G, Premaratne M (2012)

Opt Express 20:661612. Rukhlenko ID, Belov PA, Litchinitser NM, Boltasseva A (2012)

Adv OptoElectron 2012(514270)13. Zhu W, Rukhlenko ID, Premaratne M (2012) Appl Phys Lett

101:03190714. Cai W, Shalaev V (2010) Optical metamaterials: fundamentals

and applications. Springer, New York15. Mozjerin I, Gibson EA, Furlani EP, Gabitov IR, Litchinitser NM

(2010) Opt Lett 35:324016. Dalarsson M, Tassin P (2009) Opt Express 17:674717. Litchinitser NM, Maimistov AI, Gabitov IR, Sagdeev RZ, Shalaev

VM (2008) Opt Lett 33:235018. Sihvola A (2009) Eur Phys J Appl Phys 46:32602(119. Vinogradov AP, Dorofeenko AV, Zouhdi S (2008) Phys Usp

51:485

Plasmonics

20. Orlov AA, Voroshilov PM, Belov PA, Kivshar YS (2011) PhysRev B 84:045424

21. Wallen H, Kettunen H, Sihvola A (2009) Metamaterials andplasmonics: fundamentals, modelling, applications. Chap. mix-ing formulas and plasmonic composites. Springer, Netherlands,Dordrecht, pp 91–102

22. Mackay TG, Lakhtakia A (2009) Opt Commun 282:247023. Fourn C, Brosseau C (2008) Phys Rev E 77:01660324. Mackay TG, Lakhtakia A (2004) Opt Commun 234:35

25. Gollub J, Hand T, Sajuyigbe S, Mendonca S, Cummer S, SmithDR (2007) Appl Phys Lett 91:162907

26. Dykhne AM, Eksp Zh (1970) Teor Fiz 59:11027. Baskin EM, Entin MV, Sarychev AK, Snarskii AA. (1997) Phys-

ica A 242:4928. Lifshitz EM, Pitaevskii LP (2002) Physical kinetics. Elsiver

Butterworth-Heinemann, Oxford29. Landau LD, Lifshitz EM, Pitaevskii LP (1984) Electrodynamics

of continuous media. Elsiver Butterworth-Heinemann, Oxford