1
Supplementary Material
Metawaveguide for Asymmetric I
nterferometric Light-Light Switch
ing
Han Zhao
1†
, William S. Fegadolli
2†
, Jiakai Yu
1†
, Zhifeng Zhang
1
, Li Ge
3
, Axel Scherer
2
and
Liang Feng
1*
1
Department of Electrical Engi
neering, The State University of
New York at Buffalo, Buffalo,
NY 14260, USA
2
Department of Physics and Kavli Nanoscience Institute, California Institute of Technology,
Pasadena, CA 91125, USA
3
Department of Engineering Scien
ce and Physics, College of Staten
Island, CUNY, Staten Island,
NY 10314, USA and The Graduate Center
, CUNY, New York, NY 10016, USA
†
These authors contributed equally to this work.
*Email:
fengl@buffalo.edu
1. Asymmetric inteferometric light-light switching with coheren
t perfect absorption (CPA)
Under the CPA condition (
11
0
M
), it is straightforward to
show, based on the optical
transfer matrix [Eq. (1)], that
11
LR
M
trrt
in terms of the transmission coefficient
t
and
reflection coefficients
L
r
,
R
r
, which means that
L
R
trr
is needed for CPA. Previous
demonstrations of CPA
4,7,34
used systems with a mirror symmetry, and as a result, the cont
rol
beam
()
B
L
has the same intensity as the signal beam
(0)
A
. In other words,
21
1
L
M
rt
in the
previously studied systems, since
21
()
0
BL M A
when there is no reflection or scattering (
(),(0) 0
AL B
).
The goal we set to achieve, i.e., using a weak control beam
()
B
L
to bring a strong signal
beam
(0)
A
into CPA, is satisfied when
ܯ
ଶଵ
≪1
. It indicates that the reflection from the left
should be much weaker than the tr
ansmission. Again combined wit
h the CPA condition, we
know that the transmission in turn must be much weaker than the
reflection from the right.
Therefore, we have identified a necessary requirement to achiev
e our goal, which is a strongly
2
asymmetric
reflection
, i.e.,
L
R
rr
. We also note that CPA is sensitive to the relative phase of
the control beam with respect to the signal beam: if we change
the phase of
()
B
L
from
21
(0)
M
A
,
then the intensity of the scattered light gradually increases f
rom zero to a significant amount, the
maximum of which is the “on”
mode of our operation while CPA pr
ovides the “off” mode.
In our approach, we manipulate th
e spatial index-absorption mod
ulation of a photonic
waveguide in the vicinity of the
exceptional point of the scatt
ering matrix, defined by
ܵ
ൌ ቀ
ݎݐ
ݎ
ோ
ݐ
ቁ
.
The exceptional point occurs when the two scattering values
ߪ
േ
ൌേ
√
ݎ
ݎ
ோ
of the scattering
matrix become the same, which is usually achieved with a vanish
ed
ݎ
and a finite
ݎ
ோ
or vice
versa. This condition is the extreme limit of asymmetric reflec
tion, which is required by an
asymmetric CPA as mentioned above. However, if we were to achie
ve a CPA right at such an
exceptional point, then the CPA condition
L
R
trr
mentioned previously requires that the
transmission has to vanish as well (together with the other ref
lection coefficient). This is
obviously not a useful situat
ion to realize a switch, as
ݎ
ݎൌ
ோ
ൌݐൌ0
means that the device acts
an optical blackhole no matter from which side it is illuminate
d or whether there is a control
beam.
This obstacle is removed if we operate near the exceptional poi
nt instead, which still
provides a strong asymmetric reflection while maintaining a fin
ite transmission coefficient. We
further note that we do not want to operate too close to the ex
ceptional point, which would lead
to a poor transmission (
ൎ0ݐ
) and be not very useful for an
energy-efficient device. Theref
ore,
we choose to operate moderately far from the exceptional point
while still keep a good intensity
ratio between the control beam and the signal beam, which is 3
in the example given in the main
text. As we show in the manuscr
ipt, this operating principle in
a quasi-PT symmetric system
provides a convenient platform t
o realize CPA with a weak contr
olling beam.
It is worth noting that the “on”
and “off” modes here are defin
ed by the eigenstates of the
scattering matrix. Since our sys
tem is linear any input can be
decomposed into these modes to
calculate the scatte
red amplitudes.
3
2. Design of non-Hermitian modulations for asymmetric interfero
metric light-light
switching.
Within the modulated region, the coupled mode equations between
forward and
backward propagating light are derived as
01
10
d()
() ()
d
d()
() ()
d
Az
iC A z
iC B z
z
Bz
iC
A z
iC B z
z
where
1
(1 ) 8
C
,
0
2
Ci
, and
1
(1 ) 8
C
are the corresponding Fourier
coefficients, while
is the attenuation constant caused by the introduced absorptio
n and
is
the coupling coefficient between forward and backward propagati
ng modes. Then the transfer
matrix
M
in Eq. (2) reads
0
1
11
12
0
1
21
22
cosh( )
sinh( )
sinh( )
sinh( )
cosh( )
sinh( )
C
C
MLi LMi L
C
C
M
iLMLiL
,
where
222
11
0
CC
C
. The CPA condition
11
0
M
in this case is
equivalent to
11
sinh( )
i
L
CC
,
0
11
cosh( )
C
L
CC
,
with the constraint
cosh
ሺ
ܮߟ
ሻ
ଶ
െsinh
ሺ
ܮߟ
ሻ
ଶ
ൌ1
taken into consideration. We then find
ܯ
ଶଵ
is
given by
11
(1 ) (1 )
CC
in the CPA mode, and we denote it by
exp( )
i
. Here
and
are the intensity ratio of sign
al to control and the incident
phase of the control (where
we assume the incident phase of the signal is always 0), respec
tively. The intensity ratio of the
signal beam and the control beam in the CPA mode is then given
by
(1 ) (1 )
, with the
total modulation length satisfying
1
2
8
sinh
1
L
.
4
Fig. S1. Evolution of intensity ratio
(strong laser signal to weak control beam) in the CPA
mode as function of the imaginary modulation depth
. The red circle marks the parameter
2, 3
used in the experimental demonstration of asymmetric interfero
metric light-light
switching.
Fig. S1 shows the dependence of intensity ratio
on the strength of imaginary
modulation
. It can be seen that approaching the exceptional point (
1
) would allow the
power ratio being infinitely large. In other words, one can bri
ng a strong laser signal to the CPA
state with an infinitesimally weak control beam by an appropria
te phase control. In contrast,
operating far away from the exceptional point will decrease the
contrast of two i
nput intensities
and eventually result in the CPA mode with equally strong incid
ences, as demonstrated in the
prior works. For experimental d
emonstration, we chose the incid
ent power ratio contrast as
3
, which necessarily corresponds to the modulation of
2
.
3. Sample fabrication.
The fabrication starts with an SOI wafer. Periodically arranged
sinusoidal shaped combo
structures are first patterned in polymethyl methacrylate (PMMA
) by electron beam lithography
with accurate alignment, followed by electron beam evaporation
of Ge/Cr and lift-off. Then the
5
Si waveguide with cosine shaped sidewall modulations is defined
with aligned electron beam
lithography using hydrogen silsesq
uioxane resist(HSQ), followed
by dry etching with mixed
gases of SF
6
and C
4
F
8
. Finally, plasma enhanced chemi
cal vapor deposition is used to
deposit the
cladding of SiO
2
on the entire wafer.
4. Experimental evaluation of ou
tput scattering coefficients
To characterize the spectra of minimum and maximum output scatt
ering coefficients, we
used CCD camera to capture the scattered light from the four gr
ating couplers in one field of
view (Fig. 3a). First, light sca
ttered from two input grating c
ouplers (
I
1
and
I
2
) was captured at
minimum camera exposure time (100 μs). In our experiment, the i
nput power of the signal beam
entering the metawaveguide was estimated approximately 2.5 μW a
nd the control power was
approximately 0.8 μW. An exemplary image is shown in Fig. S2. B
ecause of the low exposure
time, the weaker output light scattered at two output grating c
ouplers (
O
1
and
O
2
) can hardly be
detected.
Fig. S2. Field of integration fo
r measuring input signal-refere
nce power at minimum camera
exposure time of 100 μs. White rectangular zones are the areas
of coupling gratings. In this box
area, all the signals are summed to denote signal (left:
I
1
) and reference (right:
I
2
) inputs
respectively. Dashed lines show
the layout of silicon waveguide
tapers and grating couplers.
Since light intensity is linearly proportional to the brightnes
s value on captured image as
long as no saturated exposure is reached on each pixel, the inp
ut signal and reference power can
be estimated by integrating the
pixels corresponding to their o
wn coupling gratings. The field of
integration in the characteriza
tion was circled as shown in Fig
. S1. Therefore, the signal-to-
6
reference input power ratio was determined by comparing the int
egrations of
I
1
and
I
2
. By
controlling the coupling efficiency from lensed fibers to on-ch
ip waveguide ports, we tuned the
signal-to-reference input power r
atio at 3:1, consistent with t
he design.
When recording the total
output intensity ((i.e.
O
1
and
O
2
)), we increased the exposure
time of the camera to 15 ms, such that the scattered light from
output grating couplers could be
clearly detected (see Fig. S3).
At each wavelength, the phase d
ifference between the signal and
reference inputs was controlled by the optical delay line in fr
ee space. As the output intensity
changed with respect to the phase difference, we captured image
s corresponding to minimum
and maximum total output power. Fig. S3 showed the image of max
imum output at the resonant
wavelength (i.e.
0
), corresponding to the case in
Fig. 3b. Here, the total output
s (
O
1
and
O
2
)
were obtained by integrating the light labeled in white boxes i
n Fig. S3, corresponding to the
grating couplers. It is worth
pointing out that the white boxes
for integrating
I
1
,
I
2
,
O
1
, and
O
2
in
Figs. S2 and S3 are of the same size and the relative position
to the corresponding grating
coupler.
Fig. S3. Field of integration for measuring total output power
at camera exposure time of 15 ms.
White rectangular zones show the a
reas for integration correspo
nding to two output coupling
gratings (
O
1
and
O
2
). Dashed lines represent the layout of silicon waveguide taper
and grating
couplers.
Notice that the scattering light from
I
1
and
I
2
in Fig. S3 are already saturated and thus
cannot accurately reflect the act
ual collected power. Therefore
, to obtain the correct data for
I
1
and
I
2
, we reduced the camera exposure time back to its minimum (100
μs), as shown in Fig. S2.
7
Since the value detected from th
e CCD camera increases linearly
as the camera exposure time
grows, the values of
I
1
and
I
2
corresponding to total input power were rescaled by multiplyin
g
150, with respect to the maximum exposure time of 15 ms. The ou
tput scattering coefficients (
s
Q
) were then obtained using Eq. (3
) with also considering the in
sertion loss from the directional
couplers.
The observation of no scattering
at output grating couplers at
the CPA mode (Fig. S2)
does not alter under the varied exposure time. However, with th
e wavelength detuning, the
metawaveguide is away from the resonance. In such off-resonance
conditions, the
in-phase CPA
condition cannot be reached and the two output grating couplers
inevitably scatter some power
even if they reached their minimum respectively [Figs. 4(b) and
4(c)]. In this regard, longer
exposure time (if achievable) would reveal more apparent output
s at off-resonance, which is
consistent with theoretical prediction.