Supplementary Information
Experimental demonstration of a unidirectional reflectionless parity-time metamaterial at optical
frequencies
Liang Feng
1
*
†
, Ye-Long Xu
2†
, William S. Fegadolli
1,3,4†
, Ming-Hui Lu
2
*, José E. B. Oliveira
3
,
Vilson R. Almeida
3,4
, Yan-Feng Chen
2
and Axel Scherer
1
1
Department of Electrical Engineering and Kavli
Nanoscience Institute, California Institute of
Technology, Pasadena, California 91125, USA
2
National Laboratory of Solid State Microstructures and Department of Materials Science and
Engineering, Nanjing University, Nanjing, Jiangsu 210093, China
3
Department of Electronic Engineering, Instituto Tecnológico de Aeronáutica, São José dos
Campos, São Paulo,12229-900, Brazil
4
Division of Photonics, Instituto de Estudos Avançados, São José dos Campos, São Paulo,12229-
900, Brazil
†
These authors contributed equally to this work.
*e-mail:
lfeng@caltech.edu
; luminghui@nju.edu.cn
.
Experimental demonstration of a unidirectional reflectionless
parity-time metamaterial at optical frequencies
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Derivation of Attenuation and Coupling Coefficients
To obtain analytical solutions, we average the introduced modulation
cos( )
sin( )
qz
i
qz
(
4
42
nq qz nq
q
) in the entire period as
0
exp(
)
exp(
)
qq
C
iqz
C
iqz
C
,
where
2
0
(cos( )
sin( ))
42
q
q
q
C
qz
i
qz dz i
,
2
1
(cos( )
sin( )) exp(
)
48
q
q
q
q
C
qz
i
qz
iqz dz
, and
2
1
(cos( )
sin( )) exp(
)
48
q
q
q
q
C
qz
i
qz
iqz dz
. In this form of modulation, the coupled mode
equations can be easily derived as:
0
0
()
()
()
()
()
()
q
q
dA z
iC
A z
iC
B z
dz
dB z
iC
B z
iC
A z
dz
.
At the exceptional point where
1
, the coupled mode equations can be written as
2
42
dA z
Az
dz
dB z
i Az
Bz
dz
.
Therefore the corresponding transmission and reflection coefficients are
exp
L
T
,
22
2
2
sinh
exp
2
4
f
LL
R
, and
0
b
R
. We also numerically calculated transmission
and reflection in the forward direction with different modulation lengths of the
PT
metamaterial
using FDTD simulations as shown in Fig. S1. With derived analytical formulas for transmission
and reflection, the attenuation coefficient
1
0.61 m
is determined by fitting the transmission
curve and then the coupling coefficient
1
0.49 m
is obtained by fitting the reflection curve
with the best fitted attenuation coefficient.
2
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Figure S1. Numerically calculated transmission (blue dot) and reflection (red dot) spectr
a with different modulation
lengths of
PT
metamaterial at its exceptional point. Blue and red curves denote the corresponding fits with the derived
analytical formulas.
Comparison of Proposed Passive and Typical Balanced Gain/Loss
P
T
Systems
Although the proposed
PT
system is completely passive and there is no gain to compensate the
introduced loss, the obtained
PT
characteristic and its phase evolution as a function of
are very
similar to the typical balanced
PT
systems, which can be seen from the corresponding
eigenvalues of both systems. The
S
-matrix of the typical balanced gain/loss
PT
system can be
written as
0
2
00
0
2
00
1
sinh(
)
1
1
cosh(
)
cosh(
)
1
sinh(
)
1
1
cosh(
)
cosh(
)
iL
LL
S
iL
LL
,
where
2
0
1
8
. Its eigenvalues as well as their comparisons to those in the passive system
can be found in Table S1.
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Table S1. Comparison of eigenvalues in passive and balanced
PT
systems.
Passive
PT
systems
Balanced
PT
systems
01
2
22
2
2
1
sinh(
) 1
8
'
1
1
sinh (
)
64
n
iL
s
L
is unimodular.
0
0
1
sinh(
)
cosh(
)
n
iL
s
L
is unimodular (exact
PT
phase).
1
'1
n
s
is degenerate (exceptional point).
1
n
s
is degenerate (exceptional point).
1
2
22
2
2
1
sinh(
)
1
8
'
1
1
sinh (
)
64
n
L
s
L
is non-unimodular.
2
2
1 sin(
1 )
8
cos(
1 )
8
n
L
s
L
is non-unimodular (broken
PT
phase).
It is evident that
PT
symmetry of the studied passive system is similar to the
PT
symmetric phase in the balanced gain/loss system but with an additional attenuation term
a
.
Therefore these two systems share similar underlying physics and
PT
phase evolution as a
function of
. More intuitively, similarly to unidirectional invisibility in the typical balanced
PT
system, the achieved unidirectional reflectionless effect at the exceptional point in the passive
system is also due to multiple reflections of guided light at different units and their phase
cancellation.
Broadband Response of
P
T
Characteristics
To study the broadband response of our
PT
metamaterial, frequency detuning is considered by
adding the phase mismatch term into the coupled-mode equations:
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2
2
()
1
()
()
28
()
1
()
()
28
i
kz
i
kz
dA z
Az i
Bze
dz
dB z
Bz i
Aze
dz
,
where
1
kk k
is the phase mismatch due to the frequency detuning. The transfer matrix
M
relates the complex amplitudes at two ends of
z
= 0 and
z
=
L
as follows:
11
12
21
22
( )
(0)
( )
(0)
MM
AL
A
MM
BL
B
,
where
11
cosh(
) (
) sinh(
)
2
i kL
M
L
ik
L
e
,
12
1
sinh(
)
8
i kL
M
i
Le
,
21
1
sinh(
)
8
i kL
M
i
Le
,
22
cosh(
) (
) sinh(
)
2
i kL
M
L
ik
L
e
. Therefore, it is
apparent that at the exceptional point where
1
backward reflection
2
12
22
b
M
R
M
is always 0
for any given
k
. Although the transmission and forward reflection coefficients change as a
function of
k
with the frequency detuning, the corresponding contrast ratio is 1 in a broad band
of frequencies, which is numerically confirmed as shown in Fig. S2 in next section.
Evolution of
P
T
Metamaterial
In the design, the studied
PT
metamaterial is set to have 25 periods (each period is
4
575.5 nm
q
). For the
PT
metamaterial with the original optical modulations at the
exceptional point
cos( )
sin( )
qz
i
qz
(see Fig. 1a), the reflection spectra are numerically
calculated using FDTD simulations as shown in Fig. S2a. The resonance peak in the forward
direction is located around 1550 nm, while reflection in the backward direction is almost 0 within
the entire studied range of wavelengths from 1520 nm to 1580 nm. It clearly shows the expected
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unidirectional reflectionless phenomenon corresponding to the exceptional point as the calculated
contrast ratio is always close to 1 (Fig. S2b).
Figure S2. Simulated reflection spectra of the
PT
metamaterial in both directions (
a
) and the corresponding contrast
ratio (
b
) from 1520 nm to 1580 nm.
Next, as stated in main text, to design a practical device, regions of
real
are extracted
from the original
PT
optical potentials and then the corresponding cosine modulation in the real
part is shifted
52
q
in the
z
direction to become a positive sinusoidal modulation
(
cos( )
sin(
5
2)
real
qz
qz
), while the imaginary part modulation remains at the same
location (
sin( )
sin( )
imag
i
qz
i
qz
) (Fig. S3a). Owing to the in-phase shift of the real part
modulation, the modulated phase and amplitude of guided light accumulated from both real and
imaginary part modulations remain the same after light propagates through an entire unit cell.
Therefore, these in-phase arranged modulations create an equivalent unidirectional optical
modulation similar to the original
PT
optical potentials as shown in the numerically calculated
reflection spectra for both directions (Fig. S3b). Unlike the balanced modulation in the real part in
the case of Fig. S2a, the only-positive real part modulation here results in a red-shift of the
resonance peak to the wavelength around 1560 nm. However the corresponding contrast ratio
close to 1 still manifests the expected unidirectional optical property over the studied band of
wavelengths from 1520 nm to 1580 nm (Fig. S3c). This spontaneous
PT
symmetry breaking at
the exceptional point is also visualized from mappings of light propagating inside the waveguide
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at the wavelength of 1550 nm: forward propagating light and its reflection forms strong
interference, whereas reflection in the backward direction is close to 0 (Fig. S3d).
Figure S3.
a
, designed equivalent
PT
metamaterial with separate real and imaginary part optical modulations.
b
and
c
show simulated reflection spectra in both directions and the corresponding contrast ratio from 1520 nm to 1580 nm,
respectively.
d
, simulated electric field amplitude distribution of light in a.
It is therefore concluded that the
PT
metamaterial with in-phase arranged individual real
and imaginary part optical modulations can demonstrate the unidirectional optical properties
owing to the exceptional point as well. Additional structures on top of the waveguide are then
designed accordingly to mimic its associated quantum phenomenon as stated in the main text.
Design of On-Chip Waveguide Coupler
The 3D finite difference propagation method simulation (R-Soft Design Group, Inc.) has been
implemented to design the 3 dB waveguide directional coupler that consists of two 400 nm-wide
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and 220-nm thick Si waveguides with a gap in between. Each single waveguide is a single mode
waveguide. If the gap distance between two waveguides is short enough, guided light can be
coupled between each other. As shown in Fig. S4, li
ght is first launched inside the left waveguide
and then coupled into the other that is 414 nm away due to evanescence coupling. The field
intensity gets stronger in the right waveguide as light propagates in the
z
direction. After an
interaction length of 40 μm, light intensities in two waveguides become the same (right panel of
Fig. 3S) and a 3dB waveguide coupler is thus achieved. Therefore, the final waveguide coupler is
composed of two 40 μm-long, 400 nm-wide and 220 nm-thick Si waveguides with a gap of 414
nm in between.
Figure S4. Simulated light coupling in the designed waveguide
coupler, showing evolution of electric field envelopes
(left) and intensities (right) in two waveguides as light propagates in the z direction.
Conversion from Unidirectional Reflection to Unidirectional Invisibility
Although the proposed
PT
metamaterial is completely passive and can only demonstrate
unidirectional reflectionless properties at its exceptional point, unidirectional unviability can be
simply achieved based on the presented structure by adding a suitable standard waveguide optical
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amplifier. Assuming a linear optical amplifier whose length is the same as the modulation length
of our passive
PT
metamaterial, its corresponding transfer matrix is
11
12
21
22
exp(
)
0
0
exp(
)
GG
gL
GG
g
L
,
where
g
is the corresponding gain coefficient of the added amplification waveguide. The transfer
matrix of the combined system at the exceptional point is thus expressed as
''
11
12
11
12
11
12
''
21
22
21
22
21
22
exp
0
2
sinh(
)
exp
22
2
gL
M M GG
MM
M M GG
MM
iL
g
L
.
If
g
is chosen to be
2
, this transfer matrix becomes
10
sinh(
) 1
22
iL
. Transmission
becomes unity in both directions, but reflection can only be observed in one direction, meaning
that the device become unidirectional invisible in light intensities. Additional phase due to the
added amplifier may be accumulated in transmitted light. Therefore, to get unidirectional
invisibility in both amplitude and phase, the amplifier has to be designed to support phase
accumulation of multiples of 2
π
. In practice, optical amplification waveguide can be implemented
using III/V semiconductor materials. To completely diminish reflection due to index mismatch
between III/V and Si waveguides, an adiabatically changed tapered waveguide might be
necessary to gradually convert the guided mode. However, it is in principle feasible to achieve
unidirectional invisibility by applying uniform linear optical gain on our passive structure, which
provides a much easier approach compared to the typical balanced gain/loss
PT
system.
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