This is the accepted manuscript made available via CHORUS. The article has been
published as:

PT-Symmetric Coupled-Resonator Waveguide Based on
Buried Heterostructure Nanocavities
Kenta Takata and Masaya Notomi
Phys. Rev. Applied 7, 054023 — Published 26 May 2017
DOI: 10.1103/PhysRevApplied.7.054023

PT -symmetric coupled resonator waveguide based on buried heterostructure
nanocavities
Kenta Takata and Masaya Notomi
Nanophotonics Center, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi 243-0198, Kanagawa, JapanNTT Basic Research Laboratories, NTT Corporation,
3-1 Morinosato-Wakamiya, Atsugi 243-0198, Kanagawa, Japan(Dated: April 20, 2017)
We propose and theoretically study a parity-time (PT)-symmetric photonic crystal coupled resonator optical waveguide (CROW) based on buried heterostructure nanocavities, which has potential
scalability and controllability. We analytically reveal its spectral transport properties with a tightbinding model and show the possibility of the wide-range control of its group velocity using the PT
phase transition. While the group velocity at the PT phase transition point diverges, the groupvelocity dispersion converges. A numerical estimation of the system response to temporal pulse
inputs shows that the pulse broadening is not severe in a device of hundreds of micrometers in
size. Furthermore, longer pulse duration results in a higher upper limit of the pulse peak velocity,which can be in principle superluminal. We next perform numerical simulations on the consideredphotonic crystal slab structures with the finite element method, and successfully observe PT phasetransitions. In the simulated parameter range, gain and loss coefficients of the order of 100 cm−1
meet the condition for the maximum group velocity coefficient in the context of the tight-binding
approach. 9.3-fold increase in the group velocity at 1502 nm is obtained in a three-dimensional
device by switching between the conventional and PT-symmetric CROWs. Meanwhile, we also encounter band smoothing around the phase transition, which hampers the group velocity divergence.
Our simulation result indicates that it arises from interfering evanescent waves decaying out of thedevice structure, and we discuss ways to suppress this effect.

I.

INTRODUCTION

It has been a long-standing goal to fully control the
speed of light in photonic devices. With respect toslowing the light, an array of optical microcavities withevanescent coupling, called a coupled resonator opticalwaveguide (CROW) [1, 2], is one of the most promising
platforms. In such a system, the confinement of photons in cavities and their limited tunneling give their effective mass and hence a gentle cosine-shaped photonic
band curve. As a result, a CROW can achieve an optical group velocity orders of magnitude smaller than the
speed of light in vacuum and a negligible group velocity
dispersion at the band center. Applications such as acompact optical delay line [3], enhancement of the nonlinear optical effect [4] and mode-locking [5, 6] have been
proposed. Also, the number of coherently coupled resonators in the experimental demonstration has been extended to over one thousand, which amounts to as longas 1 mm in terms of device length [7]. Here, the practicalchallenge is to significantly tune their optical responsewith external signals.
To realize optical devices with extra controllability andfunctionality a new feature in artificial dielectric structures with balanced gain and loss has recently been studied. It originates from a quantum-mechanical concept,
known as parity-time (PT) symmetry [8], in which thesystem Hamiltonian is invariant to the parity (P̂) andtime (T̂ ) reversal operation set. It has suggested an
exotic class of systems that retain their real and hence
observable energy eigenvalues, even in the presence ofpartial non-Hermitian factors such as amplification anddissipation. In addition, it is known that its protectionof quasi-Hermiticity is not always perfect, meaning that
increasing the imaginary potential can induce a phasetransition from real to imaginary eigenvalues. Becauseof the correspondence between the Schrödinger equationand the paraxial wave equation, classical optics has beensuggested as a good test bed for PT symmetry and its
symmetry breaking [9–11]. A certain condition is required for the complex refractive index n(r) = n∗ (−r),
meaning that its real part has an even spatial parity while
the imaginary part is an odd function thereby leading toPT symmetry. Various interesting phenomena have already been observed in such systems, for example, power
oscillation [9, 12], double refraction [9], unidirectional reflectivity [13, 14], and single-mode lasing [15, 16]. Here,large PT-symmetric waveguide arrays in the time domain
have been reported [12, 17], however, practical demonstrations using spatial gain and loss structures are stillchallenging in terms of scale or gain control.
PT-symmetric CROWs have both strong light confinement and optical gain and loss in their constituent
cavities. Thus, their non-Hermitian structures, alongwith the propagation direction, greatly affect their dispersion relation and hence group velocity. A theoreticalanalysis [18] shows that large systems can show significant unidirectional reflectivity and transmission restriction. Furthermore, a non-Hermitian CROW is expected
to show the divergence of its group velocity due to andexceptional point [19]. However, experimental realization of such theoretical bulk features has not been extensively pursued. Demonstrations are still limited to two
coupled high-Q microtoroid resonators [20, 21] showing

2
nonlinearity-induced isolation, which need optical pumping and fine tuning of their positions and hence coupling.
In this paper, we propose and study theoretically a new
PT-symmetric CROW based on photonic crystal buriedheterostructure mode-gap (BHM) nanocavities [22–24],
which will have both experimental scalability and gain
controllability. We first illustrates the device and describe it with a coupled mode equation corresponding
to the tight-binding model. Here, we derive the input
frequency dependences of its group velocity and groupvelocity dispersion, which are of practical importance.
They show that the group velocity diverges at the phasetransition frequency, while the group velocity dispersion
converges. This suggests that the switching from theslow light transport of a conventional CROW to fast
light transport in a PT-symmetric CROW might be realized by external pumping control. The result also clarifies that the optimum group velocity and group velocity
dispersion can be estimated by the cavity coupling andcavity spatial interval of a given conventional CROW
without gain and loss. This supports the advantage of
our system based on compact and then strongly coupling nanocavities. Next, we perform a numerical analysis of the temporal system response to pulse inputs.
It shows that the fundamental upper limit of the pulse
peak velocity depends on the input pulse width, although
the system might allow superluminal propagation. Furthermore, propagation in the device over 200 µm does
not significantly broaden input pulses with durations oftens of picoseconds. Finally, we simulate realistic device structures composed of photonic crystal nanocavities
with complex refractive indices and examine their band
structures. Here, the PT phase transition is successfully
observed assuming realistic gain coefficient values. The
simulated band curves are reproduced by a theoreticalmodel that includes up to the second-nearest neighborcoupling, where the coupling rates can be estimated withmoderate data points. However, we observed the smoothing of the exceptional point singularity [25]. It can leadto a small available group velocity and impede possible
exotic phenomena based on the PT phase transition. Our
result suggests that this is because of the interfering lightradiating out of the structure, which can be modelled bya small imaginary part of the cavity-cavity coupling [26].
Here, a high system Q factor is shown to be essential
to obtain a large group velocity. Our structure tuningachieves a 9.3-fold increase in the velocity by switching
between the conventional and PT-symmetric CROWs inthree dimension.
The remainder of this paper is organized as follows. In
Sec. II, we describe the device proposed in this study.
Sec. III shows theoretical transport properties of thesystem based on the tight-binding approach. Sec. IV
provides the result of a numerical simulation of realistic
photonic crystal structures. Sec. V discusses the tradeoff between the possible group velocity and required gain
and loss. It also mentions ways to recover the singularity
in imperfect systems. Sec. VI concludes the paper.

II. PT-SYMMETRIC CROW BASED ON
BURIED HETEROSTRUCTURE NANOCAVITIESIn this section, we describe the device structure proposed in this study. Fig. 1 (a) shows the structure
schematically. The system is composed of a semiconductor photonic crystal slab, a line defect, BHM nanocavities
along with the line defect, and an appropriate pumping
mechanism for each cavity. The photonic crystal structure is omitted from Fig. 1 (a) for simplicity. A periodic
gain and loss profile is realized by electrical or opticalpumping in BHM cavities. Fig. 1 (a) shows the case forelectrical pumping. Here, the green and purple parts inthe semiconductor slab are p-doped and n-doped, respectively, for each independent current injection channel tocontrol gain and loss in each cavity. The red and blue
rectangular BHM nanocavities mean their gain by thecurrent injection and loss by the material absorption, respectively. Closely placed nanocavities give large cavity
coupling rates, which result in the CROW bandwidth of& 1nm. Fig. 1 (b) shows the complex refractive index
profile along with the center of the line defect. Here, each
BHM works as a cavity due to a sharp index modulation
and at the same time induces gain or loss (a finite Im(n))
as a result of pumping and intrinsic material absorption.
When we use the x axis at the center of the BHM cavity array as a reference, Re(n) is even in x and Im(n) is
odd. Thus, the system guarantees the existence of PTsymmetry for suitable amounts of equal gain and loss.

FIG. 1. (a) Schematic of the proposed PT-symmetric CROW
with electrical pumping. (b) The refractive index profile alongwith the line defect and the array of BHM cavities.

3
III.

THEORETICAL ANALYSIS WITH
TIGHT-BINDING MODEL
A.

Theoretical model

For a theoretical analysis of the system, we consider asimplified coupled mode equation analysis equivalent to
the tight-binding model illustrated in Fig. 2 (a). Here,each cavity mode is described by the slowly varying electric field amplitude Em (t), where m is the cavity index.
The system comprises N pairs of cavities with alternating loss and gain, and the periodic boundary is set
by the perfect electric conductors at its edges, requiring
E0 = E2N +1 = 0. The real part of the local index foreach BHM cavity nBH is assumed to be constant, whereas
its imaginary part varies depending on whether the cavity undergoes gain (+g) or loss (−g). The cavities areevanescently coupled and their real coupling coefficient isexpressed by κ. The spatial interval of the cavities LCC
affects the transport properties estimated later. Notethat the period of identical unit cells is 2LCC , namely
a pair of cavities with gain and loss. Thus, the cosinusoidal band structure of the conventional empty CROW
(g = 0) is folded in half, as shown in Fig. 2 (b). We
obtained some device parameters such as κ = 2 × 10−3
and LCC = 2.1 µm from Ref. 3. The equation of motion
for each field amplitude can be described as a series of
temporal coupled mode equations [27, 28],
κω0
ω0
gm c
dEm
Em − i=−Em +(Em+1 + Em−1 ) .
dt
2Q
2nBH
2
(1)
Here, ω0 is the single cavity resonance frequency, whichequals the frequency of the rotating frame, Q is the cavity
quality factor and ω0 /(2Q) is the cavity decay rate. c isthe speed of light in vacuum and gm c/(2nBH ) is the gain
and loss rate in term of photons with the local phasevelocity in each cavity. We combine the first and second
terms in Eq. (1) and define the cavity index dependent
coefficient gm for Em as follows,(ω0g + nBH(m is even),
cQ
gm =
(2)
nBH ω0(m is odd).
−g + cQ
This means that the single cavity decay is just a linear
term in Eq. (1) and hence can be cancelled by the controlled gain and loss. g > 0 is the net gain (even m) andloss (odd m) coefficient for the considered cavity mode.
We also assume that a large part of the field decay works
as photonic tunneling (evanescent coupling) between thecavities. Thus, the coupling rate is sufficiently large compared to the decay rate. The device can then be regarded
as a large coherently coupled cavity system, and its total
gain and loss are balanced. Therefore, PT symmetry will
be realized without lasing as in previous studies [20, 21],
and we focus on the linear PT-symmetric system with regard to the gain and loss terms. Note that g parametrizes
various factors such as mode confinement, carrier dynamics and gain saturation.

FIG. 2. (a) Illustration of the theoretical model for the PTsymmetric CROW. (b) Sketch of the half-folded band structure of the conventional CROW (g = 0) with a spatial periodof 2LCC .

After substituting Eq. (2) into Eq. (1), we findthat the resultant field equation corresponds to the
Schrödinger equation of a non-Hermitian tight-binding
model [18],
i

dΨm
′
= igm
Ψm + κ′ (Ψm+1 + Ψm−1 ) .
dt(3)with simplification,1κ′ = κω0 ,
2

′
gm
=

(

gc
′
2nBH = g− 2ngcBH(m is even),(m is odd)

(4)

where we have changed the notation of the field Em →Ψm for the visual correspondence. With the ansatz forthe differential equation Ψm (t) = exp(−i∆ ω t)ψm , weobtain the eigenvalue equation for the frequency detuning∆ω,′ψm + κ′ (ψm+1 + ψm−1 ) .
∆ωψm = igm(5)Here, the reference for ∆ω in the rotating frame is theresonance frequency ω0 .
B.

Complex band structure with nearest-neighborcoupling

The eigenvalues of Eq. (5) can be derived by using
Bloch theorem as [18],p∆ω(Ks ) = ± 4κ′2 cos2 Ks − g ′2 ,(6)

where the eigenstates require Bloch phase factors withdiscrete values,
Ks =sπ
2N + 1(s = 1, . . . , N ).

(7)

Note that Ks is dimensionless and the wavenumber is
Ks /LCC < π/(2LCC ). The negative Ks case is omitted for simplicity. The effect of the gain and loss shows
up as −g ′2 in the square root of Eq. (6), and it enables the eigenvalues to be purely imaginary. Here, theshape of the band structure does not depend on N . We
hence focus on N → ∞, leading to band curves formedby the continuum of eigenstates (Ks → kLCC ∈ R,

4
where k is the Bloch wavenumber). Fig. 3 (a) and(b) show the real and imaginary eigenfrequency detuning ∆ω(k) for different magnitudes of the net gain andloss, g = 0, 200, 405, 500 cm−1 . The black curves givethe folded cosine-shaped band structure of the empty
CROW. When g is finite, the two real branches coalesce inside the first Brillouin zone, and non-degenerate
imaginary branches appear. The point with Re (∆ω) =Im (∆ω) = 0 is hence called the PT phase transition
point or exceptional point, and it induces switching fromextended CROW modes to a pair of amplifying and deamplifying modes localized in the gain and loss parts.
Comparing different curves, we see that as g increases
by hundreds of cm−1 the transition point moves largely
toward the inside of the band for the set of parametersconsidered. Many exotic and interesting phenomena occur around this exceptional point, and here we pay attention to the divergence of the group velocity, namely
the gradient of the real branches.

C.
FIG. 3. (a) Real and (b) imaginary parts of the eigenfrequency detuning in the PT-symmetric CROW for different
gain and loss rates g. nBH = 3.54, λ0 = c/ω0 = 1.55 µm,
κ = 2×10−3 . The curve for g = 405 cm−1 in (a) is a parabolawith the longest focal length and hence the largest group velocity coefficient, satisfying Eq. (10).

vg (∆ω) =Group velocity and group velocity dispersion

Here, we theoretically clarify the transport properties of the PT-symmetric CROW with the tight-binding
model. First, differentiation of ∆ω(k) and variable transformation give the spectral group velocity. For the lower
branch before the phase transition, the analytical expression is,

d∆ω
LCC p
d∆ω
=−
= LCC
(∆ω 2 + g ′2 ) (4κ′2 − ∆ω 2 − g ′2 ) (∆ω < 0).
dk
dKs
∆ω

Eq. (8) for different gain and loss coefficients is plottedin Fig. 4 (a). Here, ∆ω = 0 means the phase transition
point, as seen in Fig. 3 (a). Clear divergence of thecurves for finite g due to 1/∆ω → ∞ (∆ω → 0) in Eq.
(8) is observed. On the other hand, the group velocity
for the empty CROW (g = 0) is finite and small at ∆ω =0, meaning that a great change is induced in vg by the
introduction of the PT phase transition. We can easily
obtain the asymptotic behavior of vg from Eq. (8) for∆ω ≈ 0,
LCC p ′2vg → −g (4κ′2 − g ′2 )
∆ω q
LCC
2
=−
− (g ′2 − 2κ′2 ) + 4κ′4 .
(9)∆ωWe then find that the condition for the largest vg aroundthe transition point is,√(10)
g ′ = 2κ′ .
With this relation, the continuous solution of Eq. (6)for ∆ω = 0 is k = π/(4LCC ). This means that the(8)highest group velocity is obtained when the transition
occurs at the middle of the band for k > 0. The
plot for vg with g = 405 cm−1 in Fig. 3 (a) actually
meets this condition (Eq. (10)). Therefore, the curvewith g = 405 cm−1 in Fig. 4 (a) is uppermost for∆ω ≈ 0 of the four g values. The resultant coefficientin Eq. (9) with Eq. (10) is 2LCC κ′2 = LCC κ2 ω02 /2,
which is proportional to both the cavity period LCC
and the square of the coupling factor κ2 . In contrast,
the group velocity for the empty CROW at ∆ω = 0 isvg,empty = 2κ′ LCC = κω0 LCC . Thus, around the transition point (∆ω ≈ 0), vg /vg,empty ≈ κω0 /(2∆ω), showing
linear enhancement of the velocity ratio with the coupling coefficient κ. As shown previously [3], κ decays
exponentially with LCC . A smaller LCC thus leads to alarger vg for the light near the exceptional point.
Next, we examine the group velocity dispersion(GVD), which characterizes the pulse broadening. GVD
is defined as the spectral gradient of the group delay forunit propagation distance and is derived with Eq. (8),5GV D(∆ω) =

d
d∆ω



1
vg



=

−∆ω 4 − 4g ′2 κ′2 + g ′4

LCC [(∆ω 2 + g ′2 ) (4κ′2 − ∆ω 2 − g ′2 )]3/2(∆ω < 0).

(11)

mum |GV D| is 1/(2LCC κ′2 ) = 2/(LCC κ2 ω02 ), which is
the inverse of the maximum coefficient of vg for ∆ω ≈ 0.
Again, a shorter LCC leads to an exponentially larger κand hence smaller pulse broadening, depending on 1/κ2 .
Note that the curve of g = 405 cm−1 in Fig. 4 (b) shows
the smallest GVD around ∆ω = 0 when the gain andloss rates are nonzero. Overall, we can estimate the possible group velocity and group velocity dispersion in a
PT-symmetric CROW, from the device parameters of its
conventional counterpart {κ, ω0 , LCC } with no gain orloss introduced. Available values of κ′ and LCC in in-line
coupled photonic crystal nanocavities are 200 GHz and2 µm [3], while those for microtoroid resonators, which
are also good for PT-symmetric systems, can be 1 GHz
and 60 µm [20, 21]. Thus, the optimum coefficient ofvg and 1/|GV D| is by two orders of magnitude larger inphotonic crystal systems. This shows the advantage ofour device in terms of achieving fast light transport.

D.

FIG. 4. (a) The spectral group velocity and (b) group velocity
dispersion of the PT-symmetric CROW for different gain andloss rates. nBH = 3.54, λ0 = 2πc/ω0 = 1.55 µm, κ = 2 ×
10−3 . The curve for g = 405 cm−1 in (b) gives the minimum|GV D(∆ω = 0)| in finite g.

The data obtained with Eq. (11) for different g values
are displayed in Fig. 4 (b). An empty CROW (g = 0)
does not suffer from any GVD around ∆ω = 0. On theother hand, the gain and loss result in a finite GVD in
PT-symmetric CROWs. It diverges under large negative
frequency detuning (flat band regime), while it converges
at the transition point because vg (∆ω ≈ 0) ∝ 1/∆ω.
Here, the spectral width of the region with small |GV D|
depends on 4κ′2 − g ′2 . The magnitude of the GVD at thetransition point for finite g ′ can be obtained as,|GV D(∆ω = 0)| =1q. (12)LCC − (g ′2 − 2κ′2 )2 + 4κ′4

We see that the condition for the smallest |GV D(∆ω =0)| is the same as Eq. (10). The consequent mini-

Φ(m, t) =Estimation of temporal response

We have already studied analytically the group velocity
and group velocity dispersion of the system, along with
their parameter dependence. Here, we provide a numerical estimation of the temporal response of the bulk system to a pulse running over a finite distance. As shownin Fig. 3, the band structure of the system is nonlinear
and hence will lead to non-negligible high-order dispersion. Therefore, a pulse with a broad spectral distribution can show propagation slower than that with the theoretical group velocity. The dependence of the responseon the input pulse duration will hence be of significant
importance. We take these factors into account and show
the upper limit of the pulse peak velocity. In addition,
the finite length of the system limits the temporal delay of propagating optical pulses. Thus, we see whether
a detectable temporal shift of the pulses are obtainedby switching between empty and PT-symmetric CROWs
over a moderate propagation distance.
The evolving pulse amplitude at cavity site m, Φ(m, t),
is expressed by the Fourier integral of the Bloch-Floquet
eigenmodes ψ±,m (k), including their wavenumber spectral functions F± (k) and phase rotations [19, 29]. For
our system with a period of two cavities, it is given by

Z π/(2LCC )ihlmm− i∆ω+ (k) t
dk F+ (k)ψ+,m (k) exp i 2kLCC
2−π/(2LCC )Z π/(2LCC )lmmhidk F− (k)ψ−,m (k) exp i 2kLCC+− i∆ω− (k) t ,2−π/(2LCC )

(13)

6
where the subscript {+, −} corresponds to the sign of theeigenfrequency detuning ∆ω(k) (see Eq. (6)), and(1(m is even),√
ψ±,m (k) =−ig′ ± 4κ′2 cos2 (kLCC )−g′2(m is odd).
κ′ [1+exp(i 2kLCC )](14)
The absence of an offset for m and t in Eq. (13) meansthat the initial pulse is excited at the spatial and temporal references, i.e. m = t = 0.
To set the spectral function F± (k), we perform a variable transformation from an input frequency distribution,
 2

√
σt
2
F (∆ω, ∆ω0 , σt ) = 2π σt exp − (∆ω − ∆ω0 ) ,2(15)
which corresponds to a temporal Gaussian pulse with avariance σt2 and a central frequency detuning of ∆ω0 infree space. Note that F± (k) is not Gaussian because of
the nonlinear band structure of the considered system,
namely,F± (k) = F (∆ω, ∆ω0 , σt ) ·d∆ω± (k)
.
dk

(16)

Indeed, d∆ω± (k)/dk in Eq. (16) is the group velocity (Eq. (8)), diverging at the phase transition point.
However, the integration represented by Eq.(13) is finite
as long as the amplification by the states with imaginary eigenfrequencies in the broken PT phase (after the
PT symmetry breaking) is avoided. Here, we limit theanalysis to the exact PT phase (before the symmetry
breaking). This is because the states in the broken PT
phase have flat real bands (vg = 0) and curved imaginary bands, which only support monochromatic standing
waves with amplification and deamplification (see Fig.
3). Furthermore, we focus on a single propagation direction and hence the case where the states are pumped
with positive group velocities. As a result, the integral
ranges for the first and second terms of Eq. (13) canbe [−kPT , 0] and [0, kPT ], respectively, where kPT is
the phase transition point in [0, π/(2LCC )]. Note thatthe states with ∆ω+ (k) support positive group velocities
in the negative wavenumber region [−kPT , 0], althoughthat is omitted in the tight-binding analysis.
We consider detecting the temporal pulse intensities at
m = 100, corresponding to a distance of mLCC = 210 µm
from the excitation point, for both PT-symmetric andconventional empty CROWs. Fig. 5 (a) and (b) show examples of pulse propagation for different pulse intensity
widths of 10 and 100 ps, with their central frequencies
at the phase transition point. With a short input pulse,
we can achieve a large relative shift in time by switching between the PT-symmetric and empty CROWs (Fig.
5 (a)). In addition, the pulse broadening in the PTsymmetric CROW is not significant for the propagation
distance of 210 µm. However, the maximum peak velocity (ideally corresponding to the group velocity) is limited because it is affected by the broad spectral width.

FIG. 5. Examples of the propagation of pulses with the central frequency at the exceptional point and temporal durations of (a) 10 and (b) 100 ps in the PT-symmetric (in red)and conventional (blue) CROWs. m = 100. nBH = 3.54,
λ0 = 2πc/ω0 = 1.55 µm, κ = 2 × 10−3 , g = 405 cm−1 (satisfying Eq. (10)).

A longer pulse (Fig. 5 (b)) has a narrower spectrum andhence enables a faster peak velocity. However, it is moredifficult to detect the absolute time delay caused by the
propagation in the PT-symmetric CROW and the temporal pulse shift than with the conventional CROW.
Fig. 6 shows the dependence of the forward peak velocity on the central frequency for different temporal pulse
durations. The maximum peak velocity is obtained withthe central frequency at the exceptional point for boththe PT-symmetric CROW (shown as curves with markers) and the conventional empty CROW (dash-dottedline). The peak velocity in the PT-symmetric CROW
agrees with the theoretical analysis (Eq. (8)) shown by
the dotted curve for detuning approximately larger than
the spectral width of the input pulse. On the other hand,
it does not diverge around the phase transition point,∆ω = 0. This is very probably because the peak is delayed by the third and higher-order dispersion, and theeffect is more significant for pulses with broader spectral
widths and hence shorter temporal durations. However,a long pulse can show superluminal peak propagation
there, as shown by the example for 100 ps in Fig. 5 (b),which achieves 4.66 × 108 km/s. The ratios of the maximum velocity in the PT-symmetric CROW to that in
the conventional CROW for ∆ω = 0 are 9.72, 18.4, 45.4
and 91.3 for 10, 20, 50 and 100 ps pulses, respectively.
Meanwhile, as shown in the next section, the group ve-

7FIG. 6. Dependence of forward peak velocity of the pulse
on its central frequency. Lines with markers are for the PTsymmetric CROW with different input pulse widths. The
dotted line shows the theoretical group velocity for the PTsymmetric CROW. The dash-dotted line is the group velocityof the empty CROW. The parameters are the same as thosein Fig. 5.

locity around the exceptional point is decreased by theradiation loss of the device, thus a low-loss system will
also be needed to keep the controllability of the velocity.
It is noteworthy that if we excite the bands with positive and negative group velocities at the same time, apair of pulses propagating in the opposite directions areformed. This can cause a sharp decrease in the apparent
pulse velocity around the exceptional point, which might
be seen in previous work for honeycomb photonic lattices[30]. The velocity drop might also indicate the flux velocity cancellation discussed in Ref. 19. To achieve thesole advancing pulse in our simulation above, within thescope of the exact PT phase, the system needs the selectivity of the propagation direction (i.e. wavenumber)and frequency in the excitation. The way to realize thisselectivity is nontrivial and to be further explored.

IV. NUMERICAL SIMULATION FOR
PHOTONIC CRYSTAL STRUCTURES
A.

Theoretical model

In this section, we show and discuss the result of asimulation of realistic photonic crystal structures basedon the finite element method. We perform an eigenfrequency analysis of the Maxwell equation mainly on twodimensional structures to explore basic features. We employ a three-dimensional numerical simulation to confirm
the main result, compare the devices in two and three dimensions, and estimate the actual device specifications.
Fig.
7 (a) and (b) show the simulated threedimensional device structure and its two-dimensional top
view, respectively. Here, two BHM cavities with dimensions of 2.1 µm × 0.3 µm × 0.15 µm undergo loss

(Im nBH > 0) and gain (Im nBH < 0) via the imaginary parts of their input material indices (Im nBH ).
The lattice constant of the triangular photonic crystal isa = 420 nm, and the airhole radius is R = 100 nm. Sevenrows of airholes are aligned on each side of the line defect.
As shown in Fig. 7 (a), symmetric boundary planes areplaced along with the center of the line defect and on thebottom of the structure in the simulation, while Fig. 7(b) shows the entire two-dimensional unit cell for reference. The periodic boundary condition is set to enable us
to analyze the band structure of infinite pairs of cavities.
Perfectly matched layers (PMLs) are attached outside themain structure. Two widths W of the two-dimensionalline defect part, W = 0.98W0 and W = 0.85W0 are considered. Here, W is defined as the distance between theairhole rows on the√upper and lower sides of the waveguide, and W0 = 3a. We found that narrowing theline defect improves the total Q factor of the leaky lower
CROW supermode. This effect is much more significant
in three-dimensional structures than in two-dimensional
structures. For the two-dimensional system, as in Fig. 7(b), we use the effective refractive indices nSL = 2.59 forthe InP slab and Re nBH = 2.77 for the InGaAsP BHM
cavities [22]. In the three-dimensional simulation, thethickness of the slab is 250 nm, and the material indicesnSL = 3.17 and Re nBH = 3.54 are introduced. In thevertical direction, air layers are sandwiched between theslab and PMLs. Typical device parameters {Q, κ′ , LCC }obtained in our simulation are {106 , 130 GHz, 3.4 µm}
and {5 × 105 , 50 GHz, 3.4 µm} in two and three dimensions, respectively. The cavity couplings here are smaller
than that in a Si system [3]. This might diminish valuesof vg obtained in this section, compared with our previous
tight-binding analysis. Couplings in a diagonal direction[7] will be available to increase κ′ , although those are
beyond the scope of this work.
Fig. 8 shows the considered eigenmodes for the empty
CROW (Im nBH = 0) in terms of the out-of-plane component of the magnetic field Hz (x, y). Here, the groundeven mode in terms of Hz is excited and well confined in
the cavities. In the lower band eigenmode (Fig. 8 (a)),
an antinode is formed in the middle of the line defectpart, and the two BHM cavity modes are in-phase. On
the other hand, the upper band mode (Fig. 8 (b)) has
a node at the center of the line defect and out-of-phasecavity modes.
When the cavities are close, the second and higher order couplings are not negligible and the resultant band
structure verges on that of the line defect waveguide of
the BH medium. To fit the simulation data, we apply the
Rice-Mele model [29, 31] which covers up to the second
nearest neighbor coupling,ω± (k) = ω0 − 2ρ cos(2LCC k + φ)p
± −g ′2 + 2κ′2 [1 + cos(2LCC k)]. (17)Here, ω0 is the single cavity resonance frequency. ρ and φ
are the second nearest neighbor coupling rate and phase,respectively. Note that the second nearest neighbor cou-

8

FIG. 7. (a) Schematic of the simulated device structure inthree dimensions. (b) Top view of (a) without the symmetryboundary on x-z plane, which corresponds to the considered
two-dimensional structure.

FIG. 8. Hz fields of the considered supermodes for the empty
CROW (Im nBH = 0). The lower and upper modes havein-phase and out-of-phase couplings, respectively. k = 0.

pling, that is the second term in Eq. (17), just adds asmall change in vg and does not impede its divergence atthe transition point. If we assume that the data obey Eq.
(17), all the parameters can be determined without any
free fitting parameters. κ′ , ρ and φ are obtained from theeigenfrequencies for (k, Im nBH ) = (0, 0), (π/(2LCC ), 0)(empty CROW). The ratio between Im nBH and g ′ isestimated by using the data points for k = 0 and somedifferent Im nBH values.

B.

PT phase transition by loss and gain

We first investigate the complex eigenfrequency dependence on the magnitude of the gain and loss in a twodimensional device with LCC = 8a. Here, we fix theBloch wave number to the middle of the band for k > 0,i.e. k = π/(4LCC ), to find the gain and loss rates thatgive the largest group velocity around the phase transition, within the first order limits (see Sec. III C). Fig.
9 (a) shows the wavelength λ (black curve) and system
gain or loss rate (red curve), which are calculated withRe ω± (k) and Im ω± (k), respectively. Positive and negative Im ω± (k) values correspond to the system loss andgain. The markers show the result of the numerical simulation, and the curves are drawn with Eq. (17). We
have found that Eq. (17) with the calculated device parameters does not well reproduce the simulation data interms of the position of the phase transition point. We
attributed this deviation to the higher-order couplings
and then reduced κ′ by 4% to take it into account approximately within the scope of the theoretical model.
With this shift, the data points agree well with the dependence on g ′ in Eq. (17) without any other fitting
parameters. The result presents a sharp PT phase transition from real to imaginary ω± (k) around λ ≈ 1587.25
nm. Here, the material imaginary index to reach thetransition is Im nBH = ±0.0103, and the corresponding
spatial loss and gain rates are gBH = ±823 cm−1 . Thismeans that the PT phase transition for a large change
in vg will be achievable with a realistic carrier-injection
gain in III-V quantum wells [32]. Fig. 9 (b) depicts thenorm of the magnetic field for the eigenmodes after phasetransition, with Im nBH = ±0.015. We see that we canattribute the two states with loss (Im ω± (k) > 0) andgain (Im ω± (k) < 0) to the localized modes in the left
(Im nBH > 0) and right (Im nBH < 0) cavities, respectively. Note that the cavity mode profiles before and at
the phase transition are similar to those shown in Fig. 8.
It means that the localization effect gradually appears,
as the imaginary part of the eigenfrequency increases.

C.

Band structure and group index for two- andthree-dimensional structures

With the gain and loss coefficients inducing the phasetransition at k = π/(4LCC ), we next explore the complex
band structure, to examine the behavior of the PT phasetransition and possible group velocity in the wave number space of realistic systems. Fig. 10 (a) shows the band
curves in terms of the wavelength and gain or loss rate
for a two-dimensional system with W = 0.98W0 . The parameters used here are the same as those in Fig. 9. The
small deviation between the simulated data points andtheoretical curves can probably be attributed to the insufficient compensation of the effect of third and higher
nearest-neighbor couplings. We have used Im nBH =±0.0104 considering Fig. 9 (a) and successfully observeda PT phase transition around k = π/(4LCC ). It is seenthat the real band has finite slopes after the phase transition, due to hopping propagation via the second nearest
neighbor coupling. Fig. 10 (b) shows the wavelength dependent group index ng = c/vg for the two-dimensional
device. It includes finer data points around the phasetransition than in Fig. 10 (a). When ng of the PT-

9FIG. 9. (a) Dependence of the eigenwavelength (black) and system gain or loss rate (red) on the imaginary part of the cavity
refractive index |Im nBH | at the middle of the band for k > 0: k = π/(4LCC ). Markers show the simulated data points. Curves
are based on Eq. (17) with ω0 = 189.0 THz, κ′ = 124.0 GHz, ρ = 16.17 GHz, φ = 0.008730 rad, and g ′ = 1.700 × 104 Im nBH
GHz. (b) Magnitudes of the magnetic fields for the eigenmodes with Im nBH = ±0.015. LCC = 8a, in two dimensions.

symmetric CROW is minimum, the ratio of ng comparedwith the empty CROW reaches 7.2. Note that the magnitude of vg itself depends on the coupling κ′ .
We found that a narrower line defect resulted in a
faster vg , probably due to the balanced and high-Q factors of the CROW supermodes. Fig. 10 (c) and (d) show
the complex band curves and the group index aroundthe transition point for W = 0.85W0 . Here, we haveQ = 1.5 × 106 for the upper mode and 2.0 × 106 for the
lower mode with k = g = 0. The coupling rates are increased slightly due to the better field confinement, andthe entire band structure appears nearly unchanged, except for the shorter wavelengths resulting from the narrower line defect. However, a closer look with Fig. 10(d) reveals that a smaller ng of 2.69 is obtained, and thisresults in as much as 19-fold acceleration. Note that theminimum value is slightly below the two-dimensional BH
material index Re nBH = 2.77.
Fig. 11 (a) and (b) show the complex band structure and the group index of a three-dimensional device
with W = 0.85W0 . Its coupling rates, including higher
nearest-neighbor coupling rates, are smaller than thosefor two-dimensional systems because of the radiation loss
in the z direction. Thus, the data points in Fig. 11 (c)agree well with the theoretical curves assuming Eq. (17)(without any parameter adjustment). The CROW bandwidth is about 1 nm, and the magnitude of the applied
loss and gain is 334.6 cm−1 . In Fig. 11 (b), we have arelatively large minimum index of 15.91 due to the small
coupling rates. However, it shows a steep change of ng
around the phase transition and hence gives a large maximum ratio in ng of 9.3 compared with the empty CROW.

D.

Band smoothing around phase transition

As previously seen, the group velocity for the realistic photonic crystal structure does not exhibit divergent

behavior despite the fine data points under the periodic
boundary condition. This is because the band aroundthe phase transition is smoothed by the field decay fromthe system. Fig. 12 (a) and (b) depict the complex band
curves for the two-dimensional devices with 7 and 15 rowsof photonic crystal airholes, respectively. Here, the cavity spacing is larger, LCC = 10a, thus the couplings are
smaller than those in the previous two-dimensional examples. In Fig. 12 (a), the sharp phase transition is lost
due to the radiation loss and weak coupling, and the minimum ng in the PT-symmetric CROW is as large as 35.7.
At this wavelength, there is only a 3.4-fold change in ng
compared with the empty CROW. The Q factors of theupper and lower CROW supermodes for k = g = 0 are
5.0 × 104 and 8.8 × 105 , meaning there is a low-Q stateand imbalance of the loss in the supermodes. As shownin Fig. 12 (b), a stronger confinement of light in the system with more airholes restores the large vg around thetransition. Here, the minimum ng and maximum groupindex ratio are 12.7 and 10.3, respectively. Nevertheless,the improved Q factors are 2.9 × 105 and 1.4 × 108 , thus
there is still a large difference between them.
In our system, disproportionate Q factors in the supermodes, hampering a large group velocity around thetransition, stem from the interference of the evanescentfields. It means that the gain in the BHM cavities cannot
compensate for the radiation of the superposed evanescent fields to the outside the structure. This is expressedby the non-self-conjugate complex coupling terms, whichwere phenomenologically considered [25] and then confirmed theoretically and experimentally [26] in the context of coupled waveguides. We can introduce this effectby the replacement of κ′2 with κ′2 exp(iθ) in Eq. (6).
Indeed, θ = 0.02024 is used in the theoretical curves inFig. 12 (a), which agree well with the data points. Sucha non-Hermitian term breaks the unitarity and hence
the photon number conservation in the time evolutionof the effective system Hamiltonian. Thus, it can be a

10

FIG. 10. (a) Complex band structure shown as the eigenwavelength (black) and gain or loss rate (red, positive for loss, negative
for gain) in a two-dimensional device with a line defect width of 0.98 W0 . Markers indicate the simulated data. The parameters
for the theoretical curves are the same as those for Fig 9. (b) Dependence of the group index on the wavelength in the sametwo-dimensional device. (c) Band curves of a two-dimensional device with a narrower line defect W = 0.85W0 . Dashed lines:
theoretical curves with ω0 = 198.0 THz, κ′ = 134.340 GHz, ρ = −14.26 GHz, φ = 0.007432 rad and g ′ = 2.0503 × 104 Im nBH
GHz. Here, the calculated κ′ is decreased by 3.5 % to fit the simulated data points. (d) Fine data of the group index around thephase transition wavelength for a two-dimensional device with the line defect width of 0.85 W0 . The minimum index ng = 2.69
is slightly below the cavity material effective index Re nBH = 2.77.

reason for additional loss in our system and detrimental
effects. We emphasize that Ref. 26 focuses on the material absorption in the waveguide cores and surrounding
medium, but the radiation loss in our system (causedby the additional system-reservoir coupling [33]) behavesequivalently. This is implied in the experiment of Ref. 26
using scattering loss and indicated by nearly cosinusoidal
imaginary bands obtained in the case of our conventional
CROWs with low Q and no material gain or loss.
It is noteworthy that such a complex coupling component is out of the limit of the standard tight-binding
model discussed in Sec. III. However, evanescent waves
come from the cavity decay, thus this term would be
implicitly dependent on the cavity Q factor. If we assume that the complex coupling rate has the same orderof the cavity decay rate, the approximate single-cavity
Q factor√ showing significant band smoothing would beω0 /(2 κ′2 sin θ) ∼ 104 . This supports the loss magnitude of the low-Q supermodes in Fig. 12 (a).
In the supermode where the evanescent fields outside
the BH cavities are enhanced, the decay is significant.
When the evanescent fields are cancelled, however, theloss is small while the coupling is maintained. A narrow
line defect contributes to the balanced high Q factors of
the CROW modes and hence fast vg around the phasetransition, probably because it broadens the mode gap
between the CROW and photonic crystal. Note that thedefect modification does not change the band structureof the surrounding photonic crystal.
Basically, three-dimensional structures are more likely
to suffer from band smoothing than two-dimensional systems because of the loss in the vertical direction. As
shown in Fig. 11, however, narrowing the line defect inthe device largely suppresses this effect, which is significant otherwise. Our design will hence open up the possibility of the experimental realization of PT-symmetriccavity lattices maintaining sharp exceptional point behavior, which will contribute to group velocity control
and also other phenomena, such as unidirectional reflection and nonlinear optical isolation.

V.

DISCUSSION

Here, we discuss the design and setting of the PTsymmetric CROW described in this study. First, in Sec.
III we anticipated that a larger nearest neighbor coupling
rate κ would lead to a faster vg and a smaller GVD. On
the ohter hand, a larger magnitude g of gain and loss
is needed to achieve the condition for the maximum vg

11FIG. 11. (c) Band structure for a three-dimensional device
with LCC = 8a and a line defect width of 0.85 W0 . ω0 = 199.7
THz, κ′ = 46.87 GHz, ρ = −4.089 GHz, φ = −0.006088
rad, and g ′ = 1.661 × 104 Im nBH GHz for theoretical curves(dashed lines). (d) Group index versus the eigenwavelength
in the same three-dimensional device.

around the phase transition (Eq. (10)). Our simulation
result shows that the required g for the three-dimensional
devices with LCC = 2a, 3a and 4a are 460.1, 334.6 and110.4 cm−1 , respectively. These values are typical in reports for other photonic devices [15, 34], and thus will
also be realizable in our devices. However, we have todesign the cavity interval LCC with caution. A long
LCC will result in a small κ and induce significant bandsmoothing effects. On the other hand, a short LCC willcomplicate the band structure due to large higher ordercouplings, in addition to the need for large g.
Finally, we mention the literature of ways to keep thePT symmetry in realistic systems. A recent theoreticalreport [35] showed that the gain-induced dispersion incoupled GaAs waveguide cores could wash out the phasechange itself, as well as the singularity. Here, the mode
index detuning caused by the structural asymmetry andimbalanced gain and loss in the waveguides were introduced to restore their PT-symmetric behavior. As thedetuning of constituent waveguides has been examined
to dominate the imaginary part of the coupling and heal
the singularity [25], the radiation loss might conversely
work as a dispersion and detuning compensator, although
this is out of the scope of this paper. In future experi-

FIG. 12. Complex band curves for two-dimensional devices
with different numbers of rows of airholes for photonic crystals, namely different system Q factors. (a) 7 rows and (b)15 rows on both sides of the line defect. (a) shows the bandsmoothing caused by significant radiation loss, and the enhancement of vg is limited. In (b) a sharp phase transition
structure is restored by a larger Q factor. LCC = 10a and theline defect width is 0.98 W0 .

ment, slightly detuned cavity arrays might be picked forsmall system demonstrations. Delicate injection current
control, including biased pumping, will also be considered.
VI.

CONCLUSION

In conclusion, we have proposed a PT-symmetric
CROW based on buried heterostructure photonic crystal nanocavities and studied theoretically its application for controlling transport properties. Using the onedimensional tight-binding model, we derived analyticalexpressions of the frequency dependence of its group velocity and group velocity dispersion. Within the limitsof this first order approximation, the group velocity diverges while the group velocity dispersion converges at
the PT phase transition point, showing the good potential of the device. Here, the condition for the maximum velocity and the minimum group velocity dispersion around the transition point is the same. In thiscase, the velocity is enhanced and the dispersion is sup-

12pressed, depending on the square of the cavity couplingrate. An estimation of the system temporal response to
input pulses shows that pulse duration of 10 ps is shortenough to detect a boost by switching between conventional and PT-symmetric CROWs with a length of 210µm, while the ratio of their pulse peak velocities is relatively limited. Longer duration and hence a narrower
spectral distribution of the pulse provide with a higher
upper limit of the velocity around the PT phase transition point, although a judicious pulse excitation setup
could be required in experiment. In a device simulation based on the finite element method, we successfully
observed the PT phase transition in the proposed photonic crystal structures with controlled amplification andabsorption. In the parameter range we examined, thesecond nearest-neighbor coupling is not negligible andit bends the band curves after the phase transition. For
three-dimensional devices, loss and gain magnitudes withan order of 100 cm−1 are sufficient to satisfy the condition for the maximum group velocity around the phasetransition. A problem as regards achieving a large groupvelocity in the actual device is the band smoothing that
destroys the exceptional point singularity. Our simulation result indicates that the main reason for this is the
decay of the interfering evanescent waves from the pairsof cavities, which can be considered as the imaginary partof the cavity coupling rate. Increasing and balancing the
Q factors of the CROW supermodes with an appropriate
design will open up the possibility of the wide-range controllability of the optical transport, potentially including
superluminal light propagation.

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ACKNOWLEDGMENTS

We thank Akihiko Shinya, Kengo Nozaki, Masato
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work was supported by Japan Science and Technology
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