Topologically protected midgap states in complex photonic lattices

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by  Henning Schomerus

Topologically protected midgap states in complex photonic lattices
H. Schomerus
Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom
(Dated: December 11, 2013)
One of the principal goals in the design of photonic crystals is the engineering of band gaps and
defect states. Drawing on the concepts of band-structure topology, I here describe the formation
ofexponentially localized, topologically protected midgap states inphotonicsystemswith spatially
distributed gain and loss. When gain and loss are suitably arranged these states maintain their
topological protection and then acquire a selectively tunable amplification rate. This finds appli-
cations in the beam dynamics along a photonic lattice and in the lasing of quasi-one-dimensional
3 photoniccrystals.
1
0
2 Sincetheinceptionofthefield[1,2],thedesignofpho- anism to induce the midgap state in the beam propaga-
tonic crystals with band gaps and defect states has been tion through a photonic lattice; the beam can then be
n
a facilitatedbydrawinganalogiestocondensedmattersys- manipulated via adiabatic pumping of light. In an alter-
J tems. A novel impetus for such endeavors is provided native realization that includes active components, the

4 by the discovery of topological insulators and supercon- midgap state constitutes a selectively amplified mode in
ductors, systems which occur in distinct configurations a quasi-one-dimensionalphotonic crystal laser.

]
s that cannot be connected without closing a gap in the Complex Su-Schrieffer-Heeger model.—The SSH
c band structure and consequently display robust surface model was originally introduced to describe fractional-
i
t and interface states [3, 4]. Recent works have started to ized charges in polyacetylene; an exponentially localized
p
transfer concepts of band-structure topology to the pho- midgap state then forms at a defect in the dimerization
o
. tonic setting. Thus far, this has opened up avenues for pattern [14]. I consider a version (the cSSH model,
s
unidirectionaltransport[5,6],adiabaticpumpingoflight shown in Fig. 1) which applies to photonic lattices and
c
i [7], and creation of photonic Landau levels [8, 9], as well crystals and incorporates distributed loss and gain [15–
s
y as the creation of bound and edge states via dynamic 17]. The original SSH model consists of a tight-binding
h modulation in the time domain [10, 11]. chain with alternating coupling constants t and t
a b
p Thepracticalutilityoftopologicalconceptsinphoton- (for being specific let us assume t > t > 0), and a
a b
[
icswilldependmuchontherobustnessversusabsorption defect in this sequence which supports the topologically

1 and amplification. These processes do not have an elec- protected midgap state (see Fig. 1). The fundamental
v troniccounterpart;theyrendertheeffectiveHamiltonian unit cell is composed of two sites (labeled A and B)
7 non-hermitian, and break time-reversal symmetry—but with amplitudes ψ(A) and ψ(B), where the integer n
7 n n
7 in a different way than a magnetic field, whose presence enumerates the unit cells. To explore the effects of loss
0 ofabsenceentersthetopologicalcharacterizationofelec- and gain in photonic realizations I consider a staggered
. tronic band structures [12, 13]. One may therefore won- complex onsite potential iγ = iγ¯ +iγ on the A sites
1 A
0 der whether topological protection can survive the pres- and iγB = iγ¯ − iγ on the B sites. This modification
3 ence of gain and loss. defines the cSSH model. The tight-binding equations
1 Remarkably, as shown here for a complex version of read
:
v the Su-Schrieffer-Heeger (SSH) model[14], such robust-
i ness can be demonstrated for a photonic realization of εψn(A) =iγAψn(A)+t′nψn(B−)1+tnψn(B), (1a)
X
topologicallyprotectedmidgapstates,localizedatanin- εψ(B) =iγ ψ(B)+t ψ(A)+t′ ψ(A), (1b)
r n B n n n n+1 n+1
terface in the interior of the system. Under the influ-
a
ence of spatially distributed gain and loss [15–17], these where t is the intradimer coupling and t′ is the inter-
n n
states not only maintain their topologicalcharacteristics dimer coupling. The infinitely periodic system exists in
but also acquire desirable properties that do not have twoconfigurations—aconfigurationαwheret =t and
n a
an electronic analogue—the midgap states can be selec- t′ = t , and a configuration β where the values are in-
n b
tively amplified without affecting the extended states in terchanged such that t = t and t′ =t . These config-
n b n a
the system. urations are associated with Bloch Hamiltonians
The selective amplification of the midgap state can be
iγ f(−k) t +t eik (α)
utilized in beam manipulation and lasing. For instance, H(k)= A , f(k)= a b ,
the considered model can be realized as the coupled- (cid:18)f(k) iγB (cid:19) (cid:26)tb+taeik (β)
(2)
model theory of a photonic lattice with alternating lat-
delivering identical dispersion relations
tice spacings. In a setup with passive and lossy compo-
nents,themidgapstatecanberenderedlosslesswhileall
otherstatessufferidenticallosses. Thisprovidesamech- ε±(k)=iγ¯± t2a+t2b +2tatbcosk−γ2 (3)
q
2
(a) α β (e) γ = 0.3 ta γ = 0 γ = −0.3 ta
iγ iγ
A B (α) 1
n = −2 n = −1 n = 0 n = 1 n = 2 ta tb
S
y
(b) (c) (d)
2 –1
tRe ε / a-011 Im ε‾γγγAB ε0 (A,B)2|ψ|n1 (β)Sy 1
-2 0
-̟ 0 ̟ -2 -1 0 1 2 -4 -2 0 2 4 –1
k Re ε / t n –1 1 –1 1 –1 1
a
S S S
x x x
FIG. 1. (a) Complex Su-Schrieffer-Heeger (cSSH) chain with alternating couplings ta and tb as well as alternating imaginary
onsitepotentialiγA =i(γ¯+γ)andiγB =i(γ¯−γ)(describinglossorgaininthephotonicapplications). Forn<0thesystemis
in theαconfiguration while forn>0it is intheβ configuration. (b)Dispersion Reε(k)of theextendedstates, for tb =0.6ta
and γ =0.3ta. These states have Imε(k)=γ¯. (c) Dispersion in the complex eigenvalue plane, including the midgap state at
ε0 =iγA,whichformsduetothecouplingdefect. (d)AsintheoriginalSSHmodel,themidgapstateisexponentiallylocalized
around the interface and is confined to the A sublattice. (e) Topological characterization of thecSSH model. The outer curve
shows thetraceof thepseudospinvectorS(k) inthexy plane, fortheextendedstates intheupperbandε (k)of an infinitely
+
periodic chain in the α or β configuration (tb = 0.6ta and γ/ta = 0.3, 0, or −0.3). As in the original SSH model, Sz = 0, so
that the configurations can be characterized in terms of their winding number (0 in α and 1 in β). The enclosed loops show
thetraceofthefunctiong(k)whosepositioninthecomplexplanedeterminesthedirectionofS(k)accordingtoEq.(6). These
loops encircle the origin in β but not in α.
for extended states with dimensionless wavenumber k. Beam dynamics.—Let us first consider the manifesta-
In the original SSH model with γ¯ = γ = 0 this re- tion of the midgap state in the beam propagation along
sults in two bands, symmetrically arranged about ε = 0 a photonic lattice, composed of single-mode waveguides
and separated by a gap ∆ = 2(t − t ). In the cSSH as shown in Fig. 2. Experimentally, such lattices can be
a b
model these bands are shifted into the complex plane, realizedusing opticalfibers, quantum wells,or femtosec-
corresponding to decaying states if Imε < 0 and ampli- ondlaser-writingtechniques,producinginallcasesarrays
fied states if Imε > 0. However, this shift is uniform if of waveguides with a fixed cross-sectionalgeometry per-
|γ| < γ = ∆/2, which is imposed henceforward. Under pendiculartothepropagationdirectionz [20–22]. Inthis
c
this condition, all extended states experience the same setting the parameters γA and γB describe the intrinsic
overall gain (γ¯ > 0) or loss (γ¯ < 0). In the particular propagationconstantsofthewaveguides,whicharelossy
case γ¯ = 0 of balanced loss and gain, the dispersion re- if γA,B < 0 and amplifying if γA,B > 0. The couplings
mainsreal,whichcanbeexplainedbythePT symmetry takethevaluesta andtb,dependingonwhetherthespac-
σ [H(k)]∗σ =H(k) with Pauli matrix σ [18, 19]. ingbetweenthewaveguidesisaorb,respectively,andthe
x x x
Themidgapstateappearswhenthetwoconfigurations midgap state now arises from a defect in an alternating
are coupled together. In Fig. 1(a), the system is in the spacingsequence. ModeswithImε>0exponentiallyin-
α configuration for n < 0 and in the β configuration for creasealongthepropagationdirectionz whilethosewith
n ≥ 0, which results in a coupling defect in the middle Imε<0 decay.
of the sample. The spectrum consists of extended states I now set γ =0 and γ =−2γ <0, corresponding a
A B
within the two bands, plus an additional state at ε = setupwithpassiveAsitesandlossyB sites. Themidgap
0
iγ , as shown in Fig. 1(b,c). Going back to Eqs. (1), state is then lossless (ε = 0) while the extended states
A 0
thisvalueadmitsanexponentiallylocalizedsolutionwith decayuniformlyaccordingtoImε=γ¯ =−γ <0. Figure
ψA =(−t /t )−|n|andψB =0[Fig.1(d)]. Intheoriginal 2 illustrates the beam propagation in a lattice of 101
n b a n
SSHmodel the midgapstate sits atε =0 andpreserves fibers and a spacing defect in the center of the system.
0
the symmetry of the spectrum. In the cSSH model the Panel (a) depicts the arrangement of the fibers close to
midgapstatebreaksthissymmetryinawaythatdirectly the center of the sample. In panel (b), a broad wave
impacts on its amplification or decay rate—the midgap packetis fedinto the lattice witht =0.2t , γ =0.05t .
b a a
stateismorestablethantheextendedstatesifγ >0,and After a short transient the midgap state is populated
less stable if γ < 0. This has a topological origin, which and propagates without attenuation. In panel (c), the
isdiscussedattheendofthiswork. PriortothisIdiscuss light is fed into a single A fiber close to the center of the
applications of the selective amplificationmechanism for sample. Again, the midgap state is populated; it is now
the manipulation of beams and lasing. less localized because here t = 0.6t . In panel (d), the
b a
3
(a) (c) (e) (a) (b)cSSH (c)cSSH’
110200 110200 110200 β tRe ε /a-011
z t× a6800 z t× a6800 z t× a6800 α γA
40 40 40 B m ε_γ
20 20 20 A a0 I
0−10 x0 10 −050 x0 50 0−10 x0 10 γB0 20 40 60 80 1000 20 40 60 80 100
(b) (d) (f) i i
120 120 120
100 100 100 2n 16 (e)Helmholtz
z t× a468000 z t× a468000 z t× a468000 2 2ψa||050 (d) Helmholtz 0.01 01nm Re ω caRe /(/)0 14
2−0050 x0 50 2−0050 x0 50 2−0050 x0 50 −0.01 I ω caIm /(/)0 000...000123
0 0
FIG. 2. (a) Realization of the cSSH model in a photonic -20 -10 0x / a0 10 20 280 300i 320 340
lattice of single-mode waveguides with intrinsic propagation
constants γA and γB as well as alternating spacings a and FIG. 3. (a) Realization of the cSSH model in a quasi-one-
b, and a defect in that spacing sequence (around x = 0). dimensional photonic laser with a staggered arrangement of
(b) Beam propagation of an initially broad wave packet in active(A)andlossy(orpassive)components(B)inaunitcell
a lattice of 101 waveguides with tb = 0.2ta, γA = 0, γB = of size a0. (b) Spectrum of a finite system with 101 modes
−0.1ta. (c,d)BeampropagationwithlightfedintoanAorB (tb = 0.6ta, γ = 0.1ta), in increasing order of Reεi (i =
fi−b0e.1rctalo.s(eet)oAxd=iab0a,tfiocrpaulmatptiicnegwofitlhigthbt=: W0.a6vteag,uγidAe=ge0o,mγeBtr=y (0c,S1S,2H,′.m..o,d10e0l))..Th(ce)mSiadmgaepfsotratceormemplaexinsγth=em(0o.s1t+am0p.2liifi)etda
close to the center of the system. (f) Beam propagation in a state. (d)Implementationinadielectricmediumwithregions
lattice of 101 waveguides, with tb = 0.2ta, γA = 0, γB = ofrefractiveindexn=1(passive),nA =2−0.01i(gain)and
−0.2ta. [30] nB =2+0.01i(loss). Selectivelyamplifiedmidgapstatesare
predominantlylocalizedinthegainmedium. (e)Thedepicted
midgapstate(indexi=308)issituatedbetweenbands8and
9 of a system of length 40a . [31]
0
lightisfedinto aneighboringBfiber ofthe samelattice.
The beam quickly subsides as the midgap state is not
populated.
the cSSH model cannowbe interpretedas the mode fre-
Figure 2(e,f) demonstrates the feasibility of adiabatic quencies ω = ε +Ω around a large central frequency
i i
light pumping [7, 23, 24] in a lattice where the interface Ω.
gradually shifts by 5 unit cells to the right. In the tran- Panel (b) shows the spectrum of a finite cSSH chain
sient region the couplings tn and t′n interpolate linearly with 101 lattice points. For γ > 0 the midgap state is
between ta and tb, with tb = 0.2ta, γ = 0.1ta. Note selectively amplified in the time domain and thus will
that the shift of the beam is opposite to the shift of the win the mode competition. With increasing γ¯ the las-
interface. ing threshold then occurs at γ¯ = −γ, at which point
These results generalize to systems with γ 6= 0. At γ = γ¯ +γ = 0 so that ω crosses into the upper half
A A 0
fixed γ, this situation differs from the passive realization of the complex plane. For γ <0, on the other hand, the
by a z-dependent intensity scaling exp(2γ z). In active extended states win the mode competition and become
A
realizationswith γ =γ¯+γ >0>γ =γ¯−γ, |γ|<γ , lasing at γ¯ =0.
A B c
the midgap state is the only amplified state while the Amplifyingandabsorbingregionswithmatchingchar-
extended states all decay. acteristics pose an experimental challenge. In panel (c)
Laser applications.—When a system with active com- this is taken into account via an additional alternating
ponents is confined in the z direction, the midgap state real part of the onsite potential, which is equivalent to
serves as a selectively amplified lasing mode. Figure 3 setting γ to a complex value (cSSH′ model). This shifts
(a) illustrates how such a system could be realized using the real part of the midgap state’s frequency but does
anarrangementofamplifyingandabsorbing(orpassive) not affect the imaginary part, which still exceeds that
regions separated by gaps of alternating length. This of the extended states (the latter now acquire a mode
providesatopologicalrealizationofmicrolasingwithdis- dependence).
tributedgainandloss[25–28]. Whiletheunderlyingwave Panels (d,e) test the applicability of these predictions
equationissecondorderintimeorfrequency,instandard foranimplementationofthelaserinadielectricmedium
slowly-varyingenvelopeapproximationtheeigenvaluesof with refractive index n = 2−0.01i in the amplifying
A
4
partsandn =2+0.01iintheabsorbingpartsofthesys- the complex potential on this sublattice, which thus de-
B
tem. These regions have lengths a /3 and are separated termines its eigenvalue ε = iγ = iγ¯ + iγ. Notably,
0 0 A
bygaps(refractiveindexn=1)ofalternatingsizea /12 this is still consistent with the constraint ε = −ε∗ im-
0 0 0
and a /4, where a is the length of the unit cell. The plied by chirality; the specific value ε = 0 only follows
0 0 0
results apply to a system of length 40a . Midgap states in the hermitian limit of the SSH model. The extended
0
form between the lowest-lying bands that approximate statespopulate both sublattices equally,whichresults in
the continuum limit. In panels (d,e) this is illustrated Imε(k)=iγ¯. Therefore,the midgapstate is morestable
for the example of bands 8 and 9; the midgap state is thantheextendedstatesifγ >0,andlessstableifγ <0.
localized in the amplifying regions and its frequency lies
much higher up in the complex plane than those of the Conclusions.— In conclusion, photonic systems can
extended states. The results correspond well to the pre- exhibit exponentially localized, topologically protected
dictionsofthecSSHmodel,withminordeviationsmostly midgap states whose stability may be controlled via dis-
in line with the cSSH′ model. tributed loss and gain. Such states can be induced in
Topological characterization.—Finally let us discuss beam propagationthroughphotonic lattices, where they
how the particular features of the midgap state relate provideaplatformforadiabaticpumpingoflight,andin
to the topological properties of the cSSH model. As in photonic crystal lasers with inhomogeneous gain, where
the SSH model, the difference between the α and β con- they exhibit selective level amplification. Remarkably,
figuration is captured via a topological phase associated the midgap states maintain their topological protection
with the Bloch functions [3, 4, 29]. To formulate this even though the loss and gain renders the underlying
characterizationitisconvenienttowritetheeigenvectors Hamiltonian nonhermitian and breaks the time reversal
of Hamiltonian (2) as symmetry of the system. This demonstrates the utility
of topological concepts in genuinely photonic settings.
f(−k) ϕ(A)(k)
ϕ(k)=N ≡ , (4)
(cid:18)ε(k)−iγA (cid:19) (cid:18)ϕ(B)(k) (cid:19)
where N is the normalization constant. Each extended
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5
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is obtained in coupled-mode theory [20–22]. I denote by