Topologically protected midgap states in complex
photonic lattices
Henning Schomerus
Department of Physics, Lancaster University, Lancaster LA1 4YB, UK (h.schomerus@lancs.ac.uk)
Received April 16, 2013; accepted May 2, 2013;
posted May 6, 2013 (Doc. ID 188919); published May 27, 2013
One of the principal goals in the design of photonic crystals is the engineering of band gaps and defect states. Here I
describe the formation of topologically protected localized midgap states in systems with spatially distributed gain
and loss. These states can be selectively amplified, which finds applications in the beam dynamics along a photonic
lattice and in the lasing of quasi-one-dimensional photonic crystals. © 2013 Optical Society of America
OCIS codes: (130.2790) Guided waves; (140.3410) Laser resonators; (160.5293) Photonic bandgap materials.
http://dx.doi.org/10.1364/OL.38.001912
Since the inception of the field [1,2], the design of pho-
tonic crystals with band gaps and defect states has been
facilitated by drawing analogies to condensed matter sys-
tems. An impetus for such endeavors is provided by the
discovery of topological insulators and superconductors,
systems that occur in distinct configurations that cannot
be con nected without closing a gap in the band structure
and consequently display robust surface and interface
states [
3]. Recent works have started to transfer concepts
of band-structure topology to the photonic setting. Thus
far, this has opened up avenues for unidirectional trans-
port [
4,5], adiabatic pumping of light [6], and the creation
of photonic Landau levels [ 7,8], as well as the creation of
bound and edge states via dynamic modulation in the
time domain [
9,10].
The practical utility of topological concepts in photon-
ics will depend much on the robustness versus absorp-
tion and amplification. Remarkably, as shown here for
a complex version of the Su–Schrieffer–Heeger (SSH)
model [
11], such robustness can be demonstrated for a
photonic realization of topologically protected midgap
states, localized at an interface in the interior of the sys-
tem. Under the influence of spatially distributed gain and
loss [
12–14], these states not only maintain their topologi-
cal characteristics but also acquire desirable properties
—the midgap states can be selectively amplified without
affecting the extended states in the system. This sets
these states apart from conventional defect states and
can be utilized in beam manipulation and lasing.
The SSH model was originally introduced to describe
fractionalized charges in polyacetylene, where exponen-
tially localized midgap states form at defects in the dime-
rization pattern [
11]. I consider a version [the complex
Su–Schrieffer–Heeger (cSSH) model, shown in Fig.
1]
that applies to photonic lattices and crystals and incor-
porates distributed loss and gain [
12–14]. The original
SSH model consists of a tight-binding chain with alternat-
ing coupling constants t
a
and t
b
(for specificity let us as-
sume t
a
>t
b
> 0) and a defect in this sequence that
supports the topologically protected midgap state (see
Fig.
1). The fundamental unit cell is composed of two
sites (labeled A and B) with amplitudes ψ
A
n
and ψ
B
n
,
where the integer n enumerates the unit cells. The cSSH
model incorporates effects of loss and gain via a stag-
gered complex onsite potential iγ
A
i¯γ iγ on the A
sites and iγ
B
i¯γ − iγ on the B sites. The coupled-mode
equations read
εψ
A
n
iγ
A
ψ
A
n
t
0
n
ψ
B
n−1
t
n
ψ
B
n
; (1a)
εψ
B
n
iγ
B
ψ
B
n
t
n
ψ
A
n
t
0
n1
ψ
A
n1
; (1b)
where t
n
is the intradimer coupling and t
0
n
is the inter-
dimer coupling. The infinitely periodic system exists in
two configurations—a configuration α where t
n
t
a
and
t
0
n
t
b
, and a configuration β where the values are inter-
changed such that t
n
t
b
and t
0
n
t
a
. These config ura-
tions are associated with Bloch Hamiltonians
ℋk
iγ
A
f −k
f k iγ
B
;fk
t
a
t
b
e
ik
α
t
b
t
a
e
ik
β
;
(2)
delivering identical dispersion relations
ε
ki¯γ
t
2
a
t
2
b
2t
a
t
b
cos k − γ
2
q
(3)
for extended states with dimensionless wavenumber k.
Fig. 1. (a) Complex Su–Schrieffer–Heeger (cSSH) chain with
alternating couplings t
a
and t
b
and alternating imaginary onsite
potential iγ
A
i¯γ γ and iγ
B
i¯γ − γ (describing loss or
gain). For n<0 the system is in the α configuration; for n>
0 it is in the β configuration. (b) Dispersion Re εk of the ex-
tended states, for t
b
0.6t
a
and γ 0.3t
a
. These states have
Im εk¯γ. (c) Dispersion in the complex eigenvalue plane, in-
cluding the midgap state at ε
0
iγ
A
, which forms due to the
coupling defect. (d) The midgap state is exponentially localized
and confined to the A sublattice.
1912 OPTICS LETTERS / Vol. 38, No. 11 / June 1, 2013
0146-9592/13/111912-03$15.00/0 © 2013 Optical Society of America
In the original SSH mod el with ¯γ γ 0, this results
in two bands, symmetrically arranged about ε 0 and
separated by a gap Δ 2t
a
− t
b
. In the cSSH model
these bands are shifted into the complex plane, corre-
sponding to decaying states if Im ε < 0 and amplified
states if Im ε > 0. However, this shift is uniform if
jγj < γ
c
Δ∕2, which is imposed henceforward. Under
this condition, all extended states experience the same
overall gain (¯γ > 0) or loss (¯γ < 0). In the particular case
¯γ 0 of balanced loss and gain, the dispersion remains
real (which can be explained by the symmetry
σ
x
ℋk
σ
x
ℋk with Pauli matrix σ
x
[15,16]).
The midgap state appears when the two configurations
are coupled together. In Fig.
1(a), the system is in the α
configuration for n<0 and in the β configuration for
n ≥ 0, joined by a coupling defect. The spectrum
[Figs.
1(b) and 1(c)] consists of extended states from
the two bands, plus an additional state at ε
0
iγ
A
.
According to Eqs. (1), this value admits an exponentially
localized solution with ψ
A
n
−t
b
∕t
a
−jnj
and ψ
B
n
0
[Fig. 1(d)]. In the original SSH model the midgap state
sits at ε
0
0. In the cSSH model the midgap state breaks
the symmetry of the spectrum—the midgap state is more
stable than the extended states if γ > 0, and less stable
if γ < 0.
Let us first consider the manifestation of the midgap
state in the beam propagation along a photonic lattice,
composed of single-mode waveguides as shown in Fig.
2.
Experimentally, such lattices can be realized using opti-
cal fibers, quantum wells, or femtosecond laser-writing
techniques, producing in all cases arrays of waveguides
aligned along the propagation direction z [
17]. In this set-
ting the parameters γ
A
and γ
B
describe the intrinsic
propagation constants of the waveguides, which are
lossy if γ
A;B
< 0 and amplifying if γ
A;B
> 0. The couplings
take the values t
a
and t
b
, depending on whether the spac-
ing between the waveguides is a or b, respectively, and
the midgap state now arises from a defect in an alternat-
ing spacing sequence. Modes with Im ε > 0 exponentially
increase along the propagation direction z, while those
with Im ε < 0 decay.
I now set γ
A
0 and γ
B
−2γ < 0, corresponding a
setup with passive A sites and lossy B sites. The mid-
gap state is then lossless (ε
0
0), while the extended
states decay uniformly according to Im ε ¯γ −γ < 0.
Figure
2 illustrates the beam propagation in a lattice of
101 fibers and a spacing defect in the center of the system.
Figure
2(a) depicts the arrangement of the fibers close to
the center of the samp le. In Fig.
2(b), a broad wave packet
is fed into the lattice with t
b
0.2t
a
and γ 0.05t
a
. After a
short time, the midgap state is populated and propagates
without attenuation. In Fig.
2(c), the light is fed into a sin-
gle A fiber close to the center of the sample. Again, the
midgap state is populated; it is now less localized because
here I set t
b
0.6t
a
. In Fig. 2(d), the light is fed into a
neighboring B fiber of the same lattice. The beam quickly
subsides as the midgap state is not populated. Figures
2(e)
and
2(f) demonstrate the feasibility of adiabatic light
pumping [
6] in a lattice where the interface gradually
shifts by five unit cell s to the right. In the transient region
the couplings t
n
and t
0
n
interpolate linearly between t
a
and
t
b
, with t
b
0.2t
a
and γ 0.1t
a
. Note that the shift of the
beam is opposite to the displacement of the individual
waveguides.
These results generalize to systems with γ
A
≠ 0.At
fixed γ, this implies a z-dependent intensity scaling
exp2γ
A
z. When such a system is confined in the z direc-
tion, it is useful to apply the slowly varying envelope
approximation and interpret the eigenvalues of the cSSH
model as the mode frequencies ω
i
ε
i
Ω around a
large central frequency Ω. In active realizations with
γ
A
¯γ γ > 0 > γ
B
¯γ − γ, jγj < γ
c
, the midgap state
is then amplified in the time domain, while the extended
states all deca y. This provides a topological realization of
microlasing with distributed gain and loss [
18–20].
Figure 3(a) illustrates how such a system could be real-
ized using an arrangement of amplifying and absorbing
(or passive) regions separated by gaps of alternating
length. Figures
3(b) and 3(c) demonstrate the applicabil-
ity of cSSH predictions for an implementation of the laser
in a dielectric medium with refractive index n
A
2 − 0.01i in the amplifying parts and n
B
2 0.01i in
the absorbing parts of the system. Midgap states form
in the gaps between the lowest-lying bands, which is illus-
trated here for bands 8 and 9. The state is localized in the
amplifying regions, and its frequency lies much higher up
in the complex plane than those of the extended states.
Let us finally discuss how the features of the midgap
state relate to the topological properties of the cSSH
model. I write the eigenvectors of Hamiltonian (
2)as
φkN
f −k
εk − iγ
A
≡
φ
A
k
φ
B
k
; (4)
where N is the normalization constant. Each extended
state can then be associated with a pseudospin vector
Fig. 2. (a) Realization of the cSSH model in a photonic lattice
of single-mode waveguides with intrinsic propagation constants
γ
A
and γ
B
as well as alternating spacings a and b, and a defect in
that spacing sequence (around x 0). (b) Beam propagation of
an initially broad wave packet in a lattice of 101 waveguides
with t
b
0.2t
a
, γ
A
0, and γ
B
−0.1t
a
. (c), (d) Beam propaga-
tion with light fed into an A or B fiber close to x 0, for a lattice
with t
b
0.6t
a
, γ
A
0, and γ
B
−0.1t
a
. (e) Adiabatic pumping
of light: waveguide geometry close to the center of the system.
(f) Beam propagation in a lattice of 101 waveguides, with
t
b
0.2t
a
, γ
A
0, and γ
B
−0.2t
a
.
June 1, 2013 / Vol. 38, No. 11 / OPTICS LETTERS 1913
S hσ
x
; σ
y
; σ
z
i S
x
;S
y
;S
z
: (5)
From Eq. (
5), S
z
0 as long as jγj < γ
c
, while
S
x
iS
y
∝ εk − iγ
A
f k ≡ gk: (6)
In the α configuration, over a sweep through the Brillouin
zone, the function gk does not encircle the origin of the
complex plane; the pseudospin therefore librates and
traces out an arc (winding number 0, topological phase
0). In the β configuration gk encircles the origin; the
pseudospin therefore rotates and traces out a circle
(winding number 1, topological phase π). Combined with
the chiral symmetry σ
z
ℋ−k
σ
z
−ℋk, the presence
of the defect must then result in a midgap state [
21]. Due
to the localization on the A sublattice, this state pos-
sesses a fully polarized pseudospin S 0; 0; 1 and in-
herits the complex potential on this sublattice, which
thus determines its eigenvalue ε
0
iγ
A
i¯γ iγ.This
state is topologically protected against hopping disorder,
while the finite gap also provides some robustness
against onsite disorder.
In conclusion, photonic systems can exhibit exponen-
tially localized, topologically protected midgap states
whose properties are controlled via distributed loss
and gain. Such states can be induced in the beam propa-
gation through photonic lattices, where they provide a
platform for adiabatic pumping of light, and in photonic
crystal lasers with inhomogeneous gain, where they
exhibit selective level amplification. Remarkably, the
midgap states maintain their topological protection even
though the loss and gain renders the underlying Hamilto-
nian non-Hermitian and breaks the time reversal sym-
metry of the system. This demonstrates the utility of
topological concepts in genuinely photonic settings.
I gratefully acknowledge discussions with Hui Cao,
Yaron Bromberg, Ramy El-Ganainy, and Jan Wiersig.
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Fig. 3. (a) Realization of the cSSH model in a quasi-one-
dimensional photonic laser with a staggered arrangement of ac-
tive (A) and lossy components (B) in a unit cell of size a
0
.
(b) Midgap state in a dielectric medium with regions of refrac-
tive index n
A
2 − 0.01i (gain) and n
B
2 0.01i (loss). The
regions have lengths a
0
∕3 and are separated by gaps (refractive
index n 1) of alternating size a
0
∕12 and a
0
∕4 (system length
40 a
0
). The state is predominantly localized in the gain medium.
(c) Bands 8 and 9 of the system, in increasing order of Re ε
i
. The
depicted midgap state has index i 308.
1914 OPTICS LETTERS / Vol. 38, No. 11 / June 1, 2013