Light scattering in pseudo-passive media with uniformly balanced gain and loss

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by  A. Basiri

Light scattering in pseudo-passive media with uniformly balanced gain and loss
A. Basiri1, I. Vitebskiy2, T. Kottos1
1Department of Physics, Wesleyan University, Middletown, CT-06459, USA and
2Air Force Research Laboratory, Sensors Directorate, Wright Patterson AFB, OH 45433 USA
(Dated: January 12, 2015)
Weintroduceaclassofmetamaterialswithuniformlybalancedgainandlossassociatedwithcom-
plexpermittivityandpermeabilityconstants. Therefractiveindexofsuchabalancedpseudo-passive
metamaterial is real. An unbounded uniform pseudo-passive medium has transport characteristics
similar to those of its truly passive and lossless counterpart with the same real refractive index.
However, bounded pseudo-passive samples show some unexpected scattering features which can be
further emphasized by including such elements in a photonic structure.
5 PACSnumbers: 42.82.Et,42.25.Bs,11.30.Er
1
0
2 In the last decade we have witnessed unprecedented
advancements in realizing artificial materials which are
n
a specifically designed to exhibit features not found in na-
J ture. Intheelectromagneticdomain,suchmetamaterials

9 are using their structural composition in order to obtain
access of all four quadrants in the real (cid:15)−µ plane. Var-
]
s ious exotic phenomena ranging from negative refraction
c to electromagnetic cloaking and super-lensing have been
i
t actively pursued [1–8]. While these opportunities have
p
been extraordinary, optical materials exhibiting exotic
o
. values of permittivity ε and/or permeability µ are often
s
c prohibitively lossy. This is especially true for composite
i optical meta-materials. A natural solution to the prob-
s
y lemistoaddagaincomponentand,thereby,tooffsetthe FIG. 1: (color online) The electric E(z) (y-z plane) and the
h losses. A pathway to achieve this goal has been recently magnetic H(z) (x-z plane) field distribution of a resonance
p proposedinRef. [9],andsubsequentlyexploredinanum- localizedmodeλr =1.256µmforaBragggratingwithanem-
[ beddedPP-defect(green). Thegratingconsistsof20lossless
ber of publications [10–28]. This proposal capitalizes on bilayers (orange and white) at each side of the defect. Their

1 the notion of Parity-Time (PT) symmetry by utilizing index of refraction is n = 1.45 and n = 1.755 and their
1 2
v balanced gain and loss elements which are judiciously width is d = 0.2167µm and d = 0.179µm. The pseudo-
1 2
0
distributed in space such that the complex index of re- passive (PP) defect layer has index of refraction np = 3.48
06 fraction n(r) satisfies the relation n(r) = n∗(−r). This correspondingtoconstituents(cid:15)p =3.48−i0.01,µp =(cid:15)∗p while
its width is d =0.0542µm.
2 research line gave many intriguing transport properties p
0 like double refraction, unidirectional invisibility, asym-
. metric transport, CPA/ Lasers etc.
1
oscillating magnetic fields. If both ε and µ are com-
0 An alternative approach is to compensate the losses
plex, a real-value refractive index n can only be realized
5
1 withgainwhilepreservingtheuniformityofthemedium. if ε = ρ2µ∗ (where ρ is real), which is quite possible in
Thiscanbedonesimplybydopingthelossymediumwith
: media with gain. In this case n = ρ|µ| and thus one
v active elements (dies), so that the doped material would
i √ expects that an unbounded medium consisting of such
X have a real refractive index n(r) = εµ = n∗(r). The material will support traveling waves as its lossless pas-
question, however, is whether such a balanced loss/gain
r sive counterpart with the same index of refraction. We
a medium with real and uniform refractive index will au-
will therefore refer to media with real refractive index n
tomatically behave as a regular passive lossless medium. and complex ε(cid:48)(cid:48) and µ(cid:48)(cid:48) as pseudo-passive (PP) media.
This is certainly true if the permeability µ is real, in
Surprisingly, once a pseudo-passive medium is confined
whichcasethenegativecontributiontoε(cid:48)(cid:48) fromgainwill
inspace,itsscatteringpropertiesarecompletelydifferent
simply offset the positive contribution from absorption
than the ones of its passive counterpart. The difference
[29–31].
becomes even more striking when a pseudo-passive ma-
In this Letter we will concern with a different scenario terial is a component of a photonic structure. For exam-
where both ε and µ are complex i.e. ε = ε(cid:48)+iε(cid:48)(cid:48), µ = ple, the Fabry-Perrot resonances of a Bragg grating that
µ(cid:48) +iµ(cid:48)(cid:48). A typical example is plasmonic metamateri- contains pseudo-passive composite layers display sub- or
als where the effective permeability is associated with superunitarytransmittanceattheband-edges. Remark-
the electric current induced in tiny ring resonators by ably, under different circumstances, the same pseudo-
2
passive material can act as a perfect absorber, or it can Furthermore,wecanshowthatdetM=1andthust =
L
trigger lasing instability. t =t. The associated transmittance T and reflectances
R
The scattering properties of a pseudo-passive struc- R are T =|t|2 and R =|r |2.
L,R L,R L,R
ture are understood better by considering monochro- An alternative formulation of the scattering process
maticwavepropagationinanone-dimensionalscattering is provided by the scattering matrix S which connects
set-up. In this case, the steady state (TE) electric field incoming and outgoing wave amplitudes i.e.
E(z) is scalar and satisfies the Helmholtz equation
(cid:18)E− (cid:19) (cid:18) E+ (cid:19)
L =S L (5)
d2E(z) (cid:16)ω(cid:17)2 E+ E−
+n2(z) E(z)=0 (1) R R
dz2 c
ThescatteringmatrixcanbewrittenintermsoftheM-
where ω is the frequency of the field and c is the speed matrix elements as
of light in vacuum while the spatially dependent index
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:112) r t 1 −M 1
of refraction n(z)= ε(z)µ(z) is considered real for any S = L R = 21 (6)
positionz. Inthecasethatthepseudo-passivemediumis tL rR M22 det(M) M12
extendedoverthewholespace,Eq. (1)admitswavesolu-
and is useful for the theoretical analysis of lasing insta-
tionsoftheformE(z)=E+exp(iknz)+E−exp(−iknz)
bilities and perfect absorption. Below we provide some
where k =ω/c is the free space wavevector. These solu-
exampleswhichillustratetheanomalousscatteringprop-
tionsareidenticaltotheonesfoundataninfinitelossless
ertiesofaboundedstructurethatincludesaPP-medium.
passive medium with index of refraction n.
We start our analysis with the investigation of the
These similarities between a PP medium and a loss-
transportpropertiesofasinglePP-layer. Theassociated
less passive medium cease to exist once we turn to the
TM that describes the scattering process is
analysis of the associated scattering problem involving
n µ n µ
bounded structures. For simplicity we assume that the M = D(L;n )−1K( p; 0)D(L;n )K( 0; p) (7)
domain which contains the pseudo-passive structure ex- 0 n0 µp p np µ0
tends in the interval 0<z <L. We further assume that
where the matrices D and K are defined as
thescattererisembeddedinahomogeneousmediumwith
uctiononinfsoiodrfmerEiqint.dt(eo1x)boaefterteqhfueraallcetftitoonoufnntih0tyewsihc.eiac.tht,ner0fion=rgss1ima.mpTplihlceietzys,o<lwu0e- K(nnml ;µµml ) = (cid:18)12(cid:18)11+−((nnnnmmll ))((µµµµmmll )) 11+−((nnnnmm(cid:19)ll ))((µµµµmmll ))(cid:19)
exp(iknL) 0
is E = E+exp(ikz)+E−exp(−ikz) while on its right D(L;n) = (8)
L L L 0 exp(−iknL)
z > L is E = E+exp(ikz)+E−exp(−ikz). We can
R R R
relate the amplitudes of forward and backward propa-
Using Eq. (4) we evaluate the transmission and reflec-
gating waves on the left of the scattering domain with
tion amplitudes of a single PP-layer:
the amplitudes on its right via the transfer matrix M
t= 2exp[−ikL] (9)
(cid:18)ER+(cid:19) = M(cid:18)EL+(cid:19). (2) 2cos[knpL]−i(yp+y1p)sin[knpL]
ER− EL− r = r = i(yp−y1p)sin(knpL) ; r =rexp(−2ikL)
L 2cos[knpL]−i(yp+y1p)sin[knpL] R
Incaseofcompositestructuresthetransfermatrix(TM)
M is a product of transfer matrices M associated with Thesubindexpindicatesthatwerefertotheconstituents
n (cid:113)
each individual composite element i.e. M = (cid:81) M . ofthePP-layerandy ≡ (cid:15)p isitscomplexadmittance.
n n p µp
The individual TM’s are evaluated by imposing the ap- The Fabry-Perrot resonances are defined by the re-
propriate boundary conditions at the interface between quirements T = 1 and R = 0 and occur at frequencies
consequentlayers nandn+1ofthestructure. ForaTE ω = k c for which sin(k n L) = 0. This implies
FP FP FP p
mode we have that the field itself is continuous while its perfectresonanttransmissionwithnolossesandnogain,
spatial derivative satisfies the relation: regardless of the value of y and despite the fact that
p
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) the constituents are complex. Let us now calculate the
1 ∂E (z) 1 ∂E (z)
n = n+1 . (3) transportcharacteristicsofthePP-layeratω (cid:54)=ω . For
FP
µ ∂z µ ∂z
n n+1 interface example, at the middle between two consecutive Fabry-
Perrot resonances, i.e. when cos[kn L]=0, we have:
The transmission t and reflection r amplitudes for p
left and right incident waves are defined as t ≡
TEhR+e/yEaL+r,erwLr≡ittEenL−i/nEtL+eramnsdotfRth≡eETL−M/EelR−em,reRnt≡s aEsR+L/ER−. T =(cid:12)(cid:12)(cid:12)yp+21/yp(cid:12)(cid:12)(cid:12)2, R =(cid:12)(cid:12)(cid:12)(cid:12)((yypp+−11//yypp))(cid:12)(cid:12)(cid:12)(cid:12)2, (10)
tL = dMet2M2 ,tR = M122,rL =−MM2212,rR = MM1222. (4) A ≡1−T −R = 2(cid:0)(cid:12)(cid:12)yyp2p+−1y(cid:12)(cid:12)p∗2(cid:1)2 =−8(|Iymp2+[yp1]|)22 <0
3
Remarkably, the off-resonance value of A is always neg- energy, due to the PP-layer, gives A≈|E |2Im((cid:15)(cid:48)(cid:48))<0
d
ative and independent of the slab thickness L indicating and thus T +R = 1−A (cid:29) 1. The transmission and
thatitsoriginisassociatedwithsurfacescatteringrather reflectionspectrumareshowninFig. 2aandconfirmthe
than bulk scattering precesses. In fact, it is straightfor- previous expectations.
ward to show that a PP-layer will result in amplification The opposite scenario is observed in the case that
A<0,foranyfrequencyω,irrespectiveofthesignof(cid:15)(cid:48)(cid:48). the magnetic field H(z) has a maximum in the domain
Next we consider the transport properties of a loss- aroundthedefectlayer. Inthiscase T +R=1−A(cid:28)1
less Bragg grating (BG) with one PP-defect layer. We (we recall that A≈|H |2Im(µ(cid:48)(cid:48))>0) and thus we have
d
assume, without loss of generality, that (cid:15)(cid:48)(cid:48) < 0. Before attenuation of the incident light.
going on with our analysis we recall that a lossless pas- The amplification/attenuation mechanism is bet-
sive defect layer supports a resonant defect mode with a ter investigated by introducing the overall amplifica-
frequency ω lying inside the photonic band gap of the tion/absorption coefficient Θ((cid:15)(cid:48)(cid:48),ω) defined as the ratio
r
BG. This resonance mode is localized in the vicinity of of the total intensity of outgoing to incoming waves:
the defect layer and decays exponentially away from the
|E−|2+|E+|2
defect. Near ωr, the entire composite structure displays Θ((cid:15)(cid:48)(cid:48),ω)≡ |EL+|2+|ER−|2 (11)
a strong resonant transmission with T = 1 (and thus L R
R=0) due to the excitation of the localized mode. Theaboveexpressioncanbewritteninthefollowingform
Θ((cid:15)(cid:48)(cid:48),ω)= (cid:12)(cid:12)(cid:12)1+ EERL−+M12(cid:12)(cid:12)(cid:12)2+(cid:12)(cid:12)(cid:12)EERL−+ −M21(cid:12)(cid:12)(cid:12)2 (12)
|M22|2(cid:18)1+(cid:12)(cid:12)(cid:12)EER−+(cid:12)(cid:12)(cid:12)2(cid:19)
L
where we have used the fact that detM = 1. The
case Θ((cid:15)(cid:48)(cid:48),ω) > 1 indicates that an overall amplification
has been achieved at the system. The opposite limit of
Θ((cid:15)(cid:48)(cid:48),ω) < 1 corresponds to attenuation. The two ex-
treme cases of Θ((cid:15)(cid:48)(cid:48),ω ) → ∞ and Θ((cid:15)(cid:48)(cid:48) ,ω ) → 0
L L CPA CPA
indicate that the system has reached a lasing instability
or behaves as a coherent perfect absorber (CPA) respec-
tively. TheconditionfortheformercaseisM ((cid:15)(cid:48)(cid:48),ω)=
22
0 which is satisfied for some values (cid:15)(cid:48)(cid:48) =(cid:15)(cid:48)(cid:48) and ω =ω .
L L
Infact,thecomplexzerosofM correspondtothepoles
22
of the scattering matrix S, see Eq. (6). If (cid:15)(cid:48)(cid:48) =0 they lie
at the lower part of the complex-ω plane due to causal-
FIG. 2: (color online) (a) The transmittance T(ω), re-
flectance R(ω) and absorption A(ω) for the set-up of Fig. ity. As|(cid:15)(cid:48)(cid:48)|increasesthepolesmovetowardstherealaxis
1 ((cid:15)(cid:48)(cid:48) < 0). A super-unitary transmittance and reflectance andatacriticalvalue(cid:15)(cid:48)(cid:48) =(cid:15)(cid:48)(cid:48) oneofthesepolesbecomes
L
is evident in the frequency domain around the resonance fre- realω =ω , thussignifyingthetransitiontoalasingac-
L
quency. (b)Theamplification/absorptioncoefficientΘversus tion. We have confirmed in Fig. 2b that the singularity
frequency has a singular point at resonance frequency which in Θ((cid:15)(cid:48)(cid:48) = (cid:15)(cid:48)(cid:48),ω = ω ) → ∞ corresponds to the lasing
signifies a lasing action. (c) The same set-up as in Fig. 1 for L L
point as it is calculated from the analysis of the poles of
the case of CPA frequency. In this case the electric field as
the associated scattering matrix.
a minimum at the defect layer while the magentic field has
a maximum (d) The amplification/absorption coefficient Θ The other limiting case of CPA is achieved when the
versus frequency shows a zero at the CPA frequency. two terms in the numerator of Eq. (12) become simul-
taneously zero i.e. when E− = E+M and E− =
R L 21 R
A completely different scenario emerges when the de- −E+/M . These two relations are simultaneously sat-
L 12
fectlayerconsistsofaPP-medium. Wefindthattheres- isfied when M M + 1 = 0 → M M = 0 (re-
21 12 11 22
onancelocalizedmodehastransmittanceandreflectance call that detM = 1). The CPA frequency ω = ω
CPA
values which are larger (smaller) that unity in distinct is evaluated from the condition M (ω ) = 0 (the
11 CPA
contrast to the case of lossless defect layers. The super frequencies for which M (ω) = 0 are excluded since
22
(sub)-unitary transmission/reflection is related with the they correspond to lasing action). It should be stressed
distribution of the electro-magnetic field around the PP- that a CPA requires coherent incident fields which sat-
defect. Letusconsiderthecaseforwhichtheelectricfield isfy an appropriate phase and amplitude relationship
E(z) takes its maximum value (anti-nodal point) at the E− = E+M = −E+/M . An example of a CPA
R L 21 L 12
positionofthedefectlayer(seeFig. 1). Obviouslytheas- is shown in Fig. 2d.
sociatedmagneticfieldH(z)willbehavinganodalpoint The above super/sub-unitary scattering features be-
at the defect layer. Then, an estimation of the absorbed comemorepronouncedwhenoneofthecompositelayers
4
of the Bragg grating is substituted by a pseudo-passive netic field node) in the middle of each A-layer [33]. As a
medium. Specifically, the Fabry-Perrot (FP) resonances consequence, Fabry-Perrot resonances close to neighbor-
which are closer to the band edges will correspond to ing photonic band edges will have different dominating
photons with small group velocities. Therefore each in- components-electricormagnetic-oftheoscillatingfield
dividualphotonatthisfrequencyresidesinthePP-layers inside a particular layer.
for a long time. This long time interaction with the PP- InourexampleinFig. 3theA-layersconsistsofaPP-
medium leads to strong amplification/attenuation fea- medium with negative (cid:15)(cid:48)(cid:48) and positive µ(cid:48)(cid:48), which means
tures. The magnitude of the enhancement/suppression thattheFabry-Perrotresonanceswiththedominantelec-
is a growing function of the total thickness of the stack, tric field component in the A-layers will be enhanced,
asopposedtothecaseofauniformpseudo-passiveslab. while the resonances with dominant magnetic field com-
In Fig. 3 we show a typical transmission spectrum of ponentintheA-layerswillbesuppressed. Thisisexactly
such composite structure. whatweseeinFig. 3,wheretheresonanttransmissionat
everyotherphotonicbandedgeisenhanced(suppressed).
In conclusion, we have introduced a class of pseudo-
passive metamaterials for which the losses are balanced
by gain in a way that the index of refraction is real and
uniform throughout the medium. Although the light
propagation in an unbounded pseudo-passive medium
has the same characteristics as in a passive lossless
medium with the same index of refraction, their scat-
tering properties differ dramatically. When such me-
dia are incorporated in a photonic structure, it can lead
to super/sub-unity transmittance/reflectance indicating
strong absorption and amplification mechanisms for the
total structure. In many occasions these effects can turn
to a coherent perfect absorption of incident waves or to
lasing instabilities. It will be interesting to investigate
the realization of these phenomena under vectorial con-
FIG. 3: (color online) The electric E(z) (blue line at y-z ditions and in higher dimensions.
plane)andthemagneticH(z)(redlineatthex-zplane)fields Acknowledgment - This work was partially supported
oftworepresentativeFPmodeswhichresideattheupper(left by an AFOSR MURI grant FA9550-14-1-0037, by a
inset)andlower(rightinset)bandedges. Thegratingconsists
LRIR-09RY04COR,andbyaNSFECCS-1128571grant.
of50bilayers(orangeindicatesaPP-layerandwhitealossless
passive layer). In the insets we show the field profiles for the
first half of the structure, i.e. 25 layers. Left (right) inset
correspond to the FP resonance at the upper (lower) edge of
thelower(upper)band. TheconstituentsofthePP-layersare
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