1
of
17
Nonreciprocal Infrared Absorption
via Resonant
Magneto
-
optical Coupling to InAs
Komron J. Shayegan
1
, Bo Zhao
2
, Yonghwi Kim
1
, Shanhui Fan
2
, Harry A. Atwater
1
*
1
Thomas J. Watson Laboratory of Applied Physics, California Institute of Technology,
Pasadena,
California 91125
2
Department
of Electrical Engineering, Ginzton Laboratory, Stanford University,
California 94305
E
m
a
i
l
:
s
h
a
y
e
g
a
n
@
c
a
l
t
e
c
h
.
e
d
u
;
b
z
h
a
o
8
@
C
e
n
t
r
a
l
.
U
H
.
e
d
u
;
k
i
m
@
n
t
t
-
r
e
s
e
a
r
c
h
.
c
o
m
;
s
h
a
n
h
u
i
@
s
t
a
n
f
o
r
d
.
e
d
u
;
haa@caltech.edu
Abstract
Nonreciprocal elements
are
a vital
building block of electrical and optical systems.
In the infrared
regime, there is a particular interest in structures that break reciprocity
because
the
ir
thermal
absorptive
(and emissive) properties
should not obey the Kirchhoff thermal
radiation law.
In this
work,
we break time
-
reversal symmetry and reciprocity in
n
-
type doped
magneto
-
optic InAs
with
a static magnetic field
where light coupling is mediated by a guided
-
mode
-
resonator (GMR)
structure whose resonant frequency coincides with
the epsilon
-
near
-
zero (ENZ) resonance of the
doped InAs
.
Using
this s
tructure
, we observe the nonreciprocal absorptive
behavior as a function
of magnetic field and scattering angle in the infrared. Accounting for resonant and nonresonant
optical scatterin
g, we reliably model experimental results that break reciprocal absorption
relations in the infrared.
The ability to design such nonreciprocal absorbers opens an avenue to
explore devices with
unequal absorptivity and emissivity in specific channels.
2
of
17
Introduction
There has been great interest and numerous theoretical proposals for nonreciprocal
absorbers
on the grounds
that
they
do not obey
the
Kirchhoff thermal radiation law
(
1
,
2
,
3
,
4
)
.
Interest in this phenomenon stem
s
from fundamental considerations, as well as applications
ranging from infrared spectroscopy,
sensing
,
and
thermal
energy conversion efficiency
(
5
,
6
)
.
For
infrared sensing applications
,
the amount of light absorbed i
n a given channel
is something that
could be
controlled in a
nonreciprocal device, where the absorptivity can be tuned from strongly
in one channel
and weak in another
and vice
-
versa
(
7
,
8
)
.
As we review in the coming section,
nonreciprocal absorption achieved through an
external magnetic field
necessitates that the
emission
is channeled through the opposite channel.
Nonreciprocal absorption is a required
functionality to enable thermal energy harvesting exceeding the
conversion efficiency constrained
by the
Shockley
-
Queis
ser limit
(
9
)
and
reach
ing
higher
theoretical
efficiency limits, such as the
Landsberg limit
(
10
,
11
,
12
)
.
The Kirchhoff thermal radiation law
can be expressed as an equality of the absorptivity
and emissivity at a given wavelength (
휆
), polarization, and angle of incidence (
휃
!
)
(
13
,
14
,
15
)
:
훼
(
휃
!
,
휆
)
=
푒
(
휃
!
,
휆
)
(
E
1)
This
equality between the angular and spectral distributions of the emissivity and absorptivity
is a
direct consequence of reciprocity, which in a scattering context takes the form:
푟
(
휃
!
,
휆
)
=
푟
(
−
휃
!
,
휆
)
(
E
2)
where
푟
(
휃
!
,
휆
)
is the
incident
radiation from
휃
!
that is not absorbed by the absorber/emitter and
reflected
through the
−
휃
!
channel
and vice
-
versa for
푟
(
−
휃
!
,
휆
)
.
The above relationships, however,
assume that the emitter/absorber obeys Lorentz reciprocity
and does not transmit any o
f the
incident radiation
(
16
)
. In a nonreciprocal system, the equality in
e
quation
E
2 is broken; and the
Kirchho
ff law is violated
(
17
)
. The nonreciprocal behavior of the reflection directly relates to the
nonreciprocal thermal radiation.
This relation is
visualized in Figure
s
1A, B.
Achievement of nonreciprocity for
infrared radiation
is a subject of widespread
investigation. Reciprocity can be broken with linear time
-
invariant, nonlinear, and linear time
-
varying platforms
(
18
,
19
,
20
,
21
)
. The use of
magneto
-
optical materials, e.g.
,
ferrites, in basic
optical elements such as isolators and circulators has been a fundamental building block in
integrated photonics for many decades. The use of bare magneto
-
optical materials to observe free
-
space radiation
from coupled plasmon
-
phonon modes has been demonstrated, however the
wavelength regimes
are limited
to the far
-
IR and the dependence on magnetic field was not
measured
(
22
)
.
Prior experimental work on free
-
space magneto
-
optical elements has investigated
c
oupling to surface plasmon resonances at a fixed incidence angle in the Otto configuration in the
far
-
infrared regime
(
23
)
,
as well as polarization rotation in the Faraday geometry
(
24
,
25
)
.
Pairing
magneto
-
optical semiconductors with photonic crystals and
examining the magnetic field and
angular dependence in the infrared has not yet been experimentally explored.
Furthermore, t
o
date
there has not been a measurement of the interaction of thermal radiation with materials that
break
reciprocal relations
in t
he far and mid
-
infrared
,
despite an abundance of theoretical designs
reporting free
-
space propagation with magneto
-
optical elements in this wavelength regime
(
26
,
27
,
28
)
.
Here w
e experimentally demonstrate strong nonreciprocal behavior, breaking the
3
of
17
reflectivity relationship in
e
quation
E
2
over a wide range of incident angles
in an infrared thermal
photonic
absorber
that incorporates magneto
-
optic InAs with a static external magn
etic field
(
29
)
.
To apply the
external magnetic
f
ield, we use a Halbach array consisting of
three
permanent
magnets
with their
fields
oriented towards
a pole
-
piece
(Figure 1C)
(
30
)
.
The pole
-
piece is made
of a soft ferromagnetic alloy which focuses the field so that it is uniform
over
the sample
.
We
control the strength of the field
through the sample
by tuning
the length
of the gap
,
l
,
between the
two pole
-
pieces
(Figure 1C, D)
.
The
maximum field used in this paper is limited by the sample
size
,
5 mm.
Results
Experiment
Our device consists of a low
-
loss, amorphous silicon (
a
-
Si)
GMR
waveguide structure
fabricated on the surface of a 500
-
μ
m
-
thick, degenerately
n
-
type doped InAs wafer
(Figure
s
1
D
,
E
) serving as our magneto
-
optical material
resonantly
excited
in the Voigt geometry
.
The
patterned
a
-
Si
structure
on the InAs
gives a clear angle
-
dependent guided mode dispersion
relation with distinct resonances for both the zero and non
-
zero magnetic field cases. While most
of the modal field of a guided mode is confined within a low
-
loss
a
-
Si resonator, some of the
modal fiel
d penetrates into the magneto
-
optic InAs layer below, which produces the nonreciprocal
effect in the presence of a magnetic field (Figure 1
F
).
Optical characterization of InAs and
resonant infrared device design
The doping concentration of the InAs is cho
sen such that the
GMR
wavelength coincides
with the ENZ wavelength
(17.3
μ
m)
in the infrared
.
The degenerately
-
doped InAs exhibits a
Drude
-
like optical response with non
-
zero off
-
diagonal permittivity values in an applied magnetic
field
:
휀
(
휔
)
=
.
휀
""
휀
"
#
0
휀
#"
휀
##
0
0
0
휀
$$
0
(
E
3
)
휀
""
=
휀
##
=
휀
%
−
휔
&
'
(
휔
+
푖
Γ
)
휔
[
(
휔
+
푖
Γ
)
'
−
휔
(
'
]
(
E
4
)
휀
"#
=
−
휀
#"
=
푖
휔
&
'
휔
(
휔
[
(
휔
+
푖
Γ
)
'
−
휔
(
'
]
(
E
5
)
휀
$$
=
휀
%
−
휔
&
'
휔
(
휔
+
푖
Γ
)
(
E
6
)
In equations
E4
-
E6
, the plasma and cyclotron frequencies are
휔
&
=
6
푛
푒
'
/
(
푚
)
휀
*
)
and
휔
(
=
푒퐵
/
푚
)
, respectively. Four
-
point probe measurements of the InAs wafer in conjunction with
ellipsometry fits of the values
Y
and
D
,
the
complex ratio amplitude and phase responses
,
give a
relaxation rate
Γ
= 4.5 THz, a high
-
frequency limit dielectric constant
ε
%
= 12.3, a carrier
4
of
17
concentration
푛
=
1.5 x 10
18
cm
-
3
, and an effective electron mass
푚
=
0.033
푚
)
, where
푚
)
=
9.109 x 10
-
31
kg
(
31
)
. The Drude
model fit to the experimental values
is
shown in Figure 2
A
.
Furthermore, our designs targeted the
ENZ
wavelength regime
(
17.3
μ
m
)
since the off
-
diagonal
terms of the dielectric function
(
휀
"#
,
휀
#"
)
are largest relative to the on
-
diagonal terms when a
magnetic field is applied to the InAs
at this wavelength
(Figure 2
B
).
We focused on controlling the nonreciprocal response
around 17.3
μ
m
since it
corresponds to the portion of the electromagnetic spectrum exhibiting thermal radiation at or
below room temperature.
We implemented this control with a
GMR structure, which is
commonly
employ
ed to enhance absorption/reflection by critical couplin
g
in
infrared applications (
32
,
33
,
34
,
35
).
We design
ed
the
GMR structure
to critically couple to
p
-
polarized free
-
space radiation,
(i.e., electric field oscillating in the
x
-
y
plane), near
the ENZ wavelength
over a range of angles
,
with the guided resonance in the thermal emitter
. Because the resonator has been designed around
the
ENZ
wavelength
,
we see a strong detuning of a resonant peak
with a magnetic field
. If no
patterned
a
-
Si
were used
,
the plasma edge of the InAs would have no distinct resonance that
could be detuned with a magnetic fiel
d. This effect is discussed in more detail in the following
section.
The
optimal dimensions for the
GMR
structure are found based on
an
analysis of the
g
uided mode in
a
uniform
a
-
Si slab
atop
n
-
InAs
.
In this approximation of the periodic
a
-
Si
, the
uniform
a
-
Si
has a thickness
t
=
d
1
/2 +
d
2
, where
d
1
is
the
periodic
element
depth
and
d
2
is
the
supporting layer thickness
of the
a
-
Si
.
The dispersion
relation
,
w
(k
x
)
,
of the fundamental guided
mode is then found
by
solving
the equation
(
26,
36
,
37
)
:
tan
(
푘
+!
푡
)
=
푘
+!
휀
+!
[
푘
,!-
+
푘
./01
x
""
]
푘
+!
'
−
휀
+!
'
푘
,!-
푘
./01
x
""
(
E7
)
w
here
푘
푆푖
=
A
푅푒
(
휀
푆푖
)
(
휔
/
푐
)
'
−
푘
"
'
,
푘
푎푖푟
=
A
푘
"
'
−
(
휔
/
푐
)
'
,
푘
퐼푛퐴푠
=
A
푘
"
'
−
(
휔
/
푐
)
'
푅푒
(
휀
퐼푛퐴푠
)
,
x
푥푥
=
1
/
푅푒
(
휀
퐼푛퐴푠
)
,
and
k
x
is the wavevector along the direction of propagation.
In this derivation
,
we focus on the reciprocal case
to g
et
a sense of matching the
slab
waveguide
to the Drude
-
like
resonance of the InAs
,
and
neglect the off
-
diagonal terms of the InAs which are zero when there
is no applied magnetic field.
In reference 26, the
derivation for the
non
-
zero magnetic field case
shows that
w
(k
x
)
¹
w
(
-
k
x
)
and
the symmetry of the dispersion relation
is broken.
A
plot of
the symmetric
angular
dispersion for varying slab thicknesses of the
a
-
Si
waveguide is provided in
the Supplementary Materials
(
Figure S
1
)
.
In the calculations
,
we used
the
fitt
ed dielectric constant values of the
n
-
InAs
from ellipsometry.
We chose a periodicity (
L
) of 7
μ
m such that the guided mode can be folded in the light
cone and couple with the free
-
space radiation. We
fine
-
tuned
the other dimensions and
used
a
GMR structure depth
d
1
=
0.72
μ
m, and supporting layer thickness
d
2
=
2.1
μ
m. Since the
periodicity of the G
MR structure itself is much smaller than the wavelength of interest, we expect
to see only specular reflection in the considered wavelength range
(
38
)
.
We note that the
linewidth of the resonance created by the GMR structure is sensitive to both the patterned area
widths
(
L
/
2
)
and
the scattering rate of the
n
-
InAs underneath
(
39
)
.
Magnetic
-
field
-
dependent
absorption
tuning
in
magneto
-
optic GMR
structure
5
of
17
The Voigt geometry used in these experiments means that the nonreciprocal behavior of
electromagnetic radiation inside of the InAs will be confined to
p
-
polarized scattering, w
here
the
electric field oscillates perpendicular to the applied magnet
ic field (Figure 3
A
). Conversely, we
expect reciprocal behavior for
s
-
polarized scattering, as the applied magnetic field is aligned with
the oscillations in the electric field, and the cross
product of the electric and magnetic fields is
zero. The recipro
cal behavior for
s
-
polarized light is shown in Figure 3
B
, where the absorpti
vity
maxima for all magnetic field values are marked with a dashed gray line at approximately 17
μ
m.
In Figure
s
3A and
B, we emphasize the spectral shift in the peak
position of the absorptivity and
offset the data for clarity.
T
he
overall absorptiv
ity for
s
-
polarized
light does not change with
varying magnetic field.
Furthermore, w
e do not observe any detectable polarization conversion
from
p
to
s
or
s
to
p
-
polarizati
on, confirming the proper alignment of our GMR structure relative
to the optical beam path of the light (Figure
s
3
C, D
). The InAs layer is optically thick and therefor
no light is transmitted through the sample, allowing us to only measure the absorptivity
and
reflectivity.
W
e observed a
significant
redshift
of the absorptivity peak due to
the
GMR,
especially
in the positive field case (0 T to 1.2 T)
. The strong magnetic dependence of the
absorptivity spectrum clearly demonstrates the reciprocity breaking e
ffect. This nonreciprocal
effect is
observed
in a wide angular and wavelength range
.
To deconvolve the contribution coming from the
n
-
type InAs and the entire GMR
structure (
n
-
type InAs with
a
-
Si on top), we simulate
d
angular absorption spectra as a function of
magnetic field for both cases and compare
d
the simulations to our measurements (Figure 4). The
simulation
s
were performed using a finite
-
difference
,
frequency
-
domain electromagnetic
simulation tool, and
incorpo
rated
the measured InAs optical properties and
a
-
Si grating
parameters. The simulated and measured plasma edge splitting for the unpatterned InAs wafer are
in good qualitative agreement, with a broad resonance visible for the positive field surface
magneto
-
plasmon and a shorter wavelength onset (16.5
μ
m) for the negative field surface
magneto
-
plasmon (Figure
s
4
A
,
C
). The addition of the
periodic
a
-
Si
does not increase the
wavelength shift induced by the plasma
-
edge splitting into two separate surface
magneto
-
plasmons observed in the unpatterned InAs case. However
,
clear maxima are seen from the
coupled
GMR structure
that can be evaluated for multiple angles and compared to the data
, which
is
further discussed
in the following section.
Comparing the mea
sured positive field for the
GMR structure
(Figure
4
B
, red
curve
) and
the simulation (Figure
4
D
, red
curve
) we note that
both
maxima occur at a resonance centered at
17.5
μ
m
.
H
owever
,
a second shoulder resonance is present in the simulation at 16.7
μ
m which is
not easily visible in the experiment. The shoulder resonance is much weaker in the experiment
likely because of the
larger material loss in both the
a
-
Si and InAs. Going from zer
o to negative
applied magnetic field at a
q
i
=
50
°
we did not see a large shift in the resonant peak position for
either the simulation or the measurement
,
but do see the resonance narrowing for the negative
field case. The simulated and experimental traces for more angles of incidence and an
intermediary 0.8 T field are included in the
S
upplementary
M
aterial
(Figure
s
S
2
-
4
).
To further highlight the effect o
f the GMR structure on nonreciprocal absorption, we look
at the intensity effect on the absorption resonance at more oblique angles. Taking
q
i
= 7
0
°
, we can
subtract the absorption of the bare InAs
around the plasma shoulder
from the GMR structure
absorpt
ion and fit a Lorentzian to the difference (Figure 5A).
When the magnetic field is then
turned on, the resonant intensity peak can be drastically tuned from absorption that greatly
exceeds the plasma shoulder (positive field) to
well below
(negative field).
Angle
-
dependence of
nonreciprocal absorption behavior
6
of
17
To more comprehensively understand and characterize the nonreciprocal response of the
hybrid GMR structure, we compare the experimental and simulated absorptivity maxima for
p
-
polarize
d light over a range of incidence angles for
퐵
= 0 T
, 0.8 T,
and 1.2 T.
To map our
dispersion across positive and negative angles, we confirm that Onsager
-
Casimir
relations hold
(Figure S
2
).
When
퐵
= 0 T
,
reciprocity is
preserved,
and we do not expect to see any reciprocity
-
breaking behavior. Consequently, the angular dispersion relation of the guided mode resonance at
the plasma edge of the InAs remains symmetric. Figure
6
A
shows the symmetric case measured
in our experiment and t
he corresponding absorptivity maxima from simulations (Figure
6
D
).
We
also observe a narrowing of the resonant linewidth for more oblique angles of incidence in both
the simulated and experimental data when no magnetic field is applied, as shown by the ver
tical
bars marking the full
-
width
-
at
-
5% maximum of each absorptivity spectrum.
When the in
-
plane
magnetic field is applied, reciprocity is broken, and the measured dispersion of the absorptivity
maxima become
s
asymmetric (Figure
s
6
B, C
). The degree of reci
procity breaking grows
at
small
er angles
(approaching
q
i
= 4
5
°
)
, which is expected from the
absorptivity maxima
found
from
the
simulation
s
(Figure
6
E
, F
).
We also observe a clear detuning of the absorptivity maxima with an increase in the
magnetic field. For both the experimental (Figure
6
B) and simulated (Figure
6
E) absorptivity
maxima at the intermediate 0.8 T magnetic field, the lineshape
width for positi
ve angles of
incidence grows considerably. This makes the detuning of the absorptivity maxima at narrow
incidence angles difficult to resolve.
Increasing
the magnetic field to 1.2 T
separates the maxima
at narrow angles such that the 5% bars
do not
overlap.
Unfortunately, our measurement system only allows for incidence angles down to 3
5
°
,
limiting the ability to compare experiments and simulations for near
-
normal incidence. However,
simulations indicate that the difference for positive and negative
incidence (positive and negative
field) decrease and approach zero as the scattering geometry approaches normal incidence since
the first
-
order magneto
-
optic effects go to zero
(Figure
6G
-
I
).
Interestingly, the measured maxima for negative incidence angles
do not shift below the
zero magnetic field maxima for oblique angles. This
contrasts with
the maxima extracted from
simulations, which redshift for oblique (
-
65
°
and
-
70
°
)
angles of incidence
.
Discussion
In
this study
,
we use
a degenerately
-
doped
magneto
-
optical semiconductor paired with a
GMR
couple
d
to free
-
space incident radiation to demonstrate
nonreciprocal absorption
in the
infrared regime when a moderate magnetic field (
퐵
up to
1.2 T) bias is applied. The degree of
reciprocity breaking is
largest at narrower
incidence angles
q
i
(45
°
-
55
°
)
,
making the design
potentially useful for cascading multiple emitters/absorbers
to achieve directional flow of energy
(
40
)
. The methods used to model, design, and fabricate the structure presented in this
manuscript
can be u
tiliz
ed for future implementations of nonreciprocal absorbers. As a future direction for
engineering higher efficiency thermal radiators, direct measureme
nt of the emissivity could be
employed to demonstrate
applications of systems that
violat
e
the Kirchhoff thermal radiation law
.
Higher quality factors of the structure could be obtained using a thinner magneto
-
optic layer with
a back reflector or a lower d
oping concentration of InAs; however
,
this would result in a longer
working wavelength. A higher mobility magneto
-
optic material, such as InSb, could also be used
to achieve a narrower linewidth
(
41
).
The
easily
identifiable
absorpti
vity
peaks at the plasma edge
are a consequence of using a periodic
a
-
Si
structure supporting a
GMR
, while the nonreciprocal
behavior comes from the plasma
-
edge splitting of the magneto
-
optic
n
-
type InAs. The
temperature dependence of the optical properties of
the
n
-
InAs are included in the
S
upplementary
7
of
17
M
aterial
as a reference for designing a
t
elevated temperatures to measure emissivity (Figure S
5
).
Beyond demonstrating
nonreciprocal absorption in the infrared
, hybrid magneto
-
optic and
photonic crystal structu
res like those reported here are of potential interest for thermal radiation
control and free
-
space information processing owing to the polarization
-
dependent response and
low required magnetic field
s
for resonance tunability.
Materials and Methods
Device Fabrication
Our fabrication of the GMR structure started by deposition of
a
-
Si on the InAs wafer
using plasma
-
enhanced chemical vapor deposition (PECVD).
The
deposition was
carried out
at
a
temperature of
200
°C
and
pressure of
800 mTorr
with a flow rate of 250 SCCM (5% SiH
4
/Ar)
for 90 seconds.
We then spin
-
coat
ed
500 nm of
ZEP 520A resist
(1 minute at 2,000 rpm)
and
bake
d
the sample for 5 minutes at 180
°C
on a hot plate
.
Electron
beam lithography
was then used to write the desired pattern into the resist, using
a beam current of 100 nA with a 300
μ
m aperture and a dose of 240
μ
C/cm
2
. After writing the
pattern into the resist, the sample was dipped in ZED N
-
50 for 2 minutes and 30 seconds for
development.
Following
exposure, the sample was baked at 140
°C for 3 minutes.
The
a
-
Si pattern
was
subsequently etched using
inductively coupled plasma
-
enhanced
reactive ion etching
(ICP RIE)
.
The etching recipe began with an O
2
followed by a SF
6
cleaning
cycle for 10 minutes each followed by 2 minutes and 30 seconds of etching with SF
6
as the
etchant gas. The same
cleaning cycle was then repeated in reverse. The sample was left in PG
remover overnight and then checked with a confocal microscope to ensure the resist had been
removed.
SEM
images
we
re then taken
at normal and 45
° tilted incidence
to confirm the GMR
str
ucture dimensions.
For the SEM images shown in this paper, the microscope was operated at an
accelerating voltage of 5 kV.
Measurements
To probe the sample, we use
d
a two
-
theta stage with a silicon carbide Globar source
and
focusing optics, exciting the sa
mple at an incident angle
휃
!
relative to the
x
-
z
plane and collect the
specular
reflected light at
−
휃
!
.
For the zero and low (up to but not including 0.8 T
)
magnetic field
measurements,
the original sample holder of the system (J. A. Woollam I.R. VASE Mark II) was
used.
For the higher magnetic field measurements,
we used
a separate adapter to accommodate
the Halbach array.
The lower magnetic field measurements are obtained using a special
ly
-
designed
aluminum
holder to separate the two neodymium magnets while holding the sample in
between the two magnets’ poles.
The field strength was tuned by inserting aluminum spacers to
increase the gap between the magnets.
We use
d
a Halbach array magnet
configuration with
tunable pole pieces to apply
0.8
<
퐵
<
1.2 T
(
42
)
. An image of the magnet assembly can be found
in the S
upplementary Material
, Figure S
7
. We measure
d
the magnetic field
strength and
uniformity across the sample surface
with a transverse
H
all sensor made from InAs (Lakeshore
HGT 1010).
The sample
wa
s in the Voigt geometry, with an in
-
plane magnetic field,
퐵
, breaking time
-
reversal symmetry. By changing the sign of the ma
gnetic field, we produce
d
the same effect as
keeping the sign of magnetic field constant while switching the positions of the source and
8
of
17
detector
(
43
,
44
,
45
)
.
The use of
linear
polarizers at both the source output and detector input
allow for a deconvolution of the physical effects
(e.g. ensure that there is no cross
-
polarization
due to sample misalignment).
Device simulations
We used the COMSOL
electromagnetic waves, frequency
domain
package
to simulate the
optical response of the GMR structure.
For
the material properties and geometries of the structure,
we used the measured values of the
a
-
Si periodicity and depth from SEM images and
n
-
InAs
optical properties from the model f
its of ellipsometry data
and four
-
point probe measurements
.
For each
trace
at a fixed angle of incidence, we swept the wavelength from 16 to 19
μ
m
in 12
-
nm
intervals.
The absorptivity was found by subtracting the reflectivity from 1, as there was no
transmission for the structure. This was verified by taking transmission measurements of the
n
-
InAs wafer.
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Acknowledgments
General:
YK used the Kavli Nanoscience Institute (KNI) at Caltech for fabrication facilities. KS would like
to acknowledge Hamid Akbari, Dennis Howard Drew, and Arne Laucht for discussions around
the design of the experiment and application of the magnetic field.
Funding:
This work was supported by DARPA NLM (KS, BZ, YK, SF, HAA). KS would like to thank the
support
by
the NSF GRFP
for a Graduate Research Fellowship.
Author contributions:
The manuscript was written through contributions of all authors. All authors have given approval
to the final version of the manuscript.
Measurements were done by KS. Fabrication was done by
YK.
The theoretical and modeling work was led by BZ and supported
by KS.
Competing interests:
The authors
declare that they have no competing interests
.
Data and materials availability:
All data discussed in this manuscript is presented and the
analysis described.
Additional data is available upon reasonable request.
11
of
17
Figures
and Tables
Fig. 1
:
Overview of nonreciprocal absorber/emitter theory, design, and implementation
.
Fig. 2: Drude reflectance and dielectric constant of degenerately doped
n
-
InAs
.
Fig. 3: Polarization dependent reflection measurements for vary
ing applied magnetic field.
Fig. 4: Magnetic field effect on
p
-
polarized absorptivity
spectra
for bare InAs and the GMR
structure
at
휃
!
= 50
°
.
Fig. 5:
Intensity effect of GMR on nonreciprocal absorption at
휃
!
=
7
0
°
.
Fig.
6
: Magnetic field strength and angular dependence of reciprocity breaking.
Supplementary Materials
Fig. S
1
:
Slab waveguide
angular
dispersion for varying
a
-
Si layer thicknesses on top of
n
-
InAs
.
Fig. S2:
Schematic of reflection setup and confirmation of the Onsager reciprocal relations
.
Fig
.
S3:
Simulated and experimental data on the narrow
-
angle transition from strong
-
to
-
weak
nonreciprocal absorption.
Fig
.
S
4
:
Experimental and simulated spectra for the absorptivity at varying incident angles and
magnetic field strengths.
Fig. S5
:
Temperature dependence of the dielectric constant of InAs.
Fig. S6: Electric field intensity plots showing the plasmon mode conf
inement within the GMR
structure.
Figure S7: Image and schematic of the Halbach array with tunable supermendur pole pieces.
12
of
17
Fig. 1
: Overview of nonrecip
rocal absorber/emitter
theory, design, and implementation.
(A
&
B)
Reciprocal and nonreciprocal relations for an absorber/emitter.
(C)
Schematic of the
Halbach array
used in the measurement.
The
poles of the magnets on each side of the system are
rotated
by
90
°
relative to one another to
increase the magnitude of
the magnetic field through the
pole pieces (gray).
The pole pieces
provide a focused and uniform magnetic field across the
sample, which is shown in the gap.
We tune the magnetic field strength by changing the gap
length
,
l
, between the pole pieces.
(
D
)
S
chematic of the measurement scheme. A silicon carbide
Globar is used as the thermal source inside of a
Fourier transform infrared (
FTIR
) spectrometer
.
The
sample is mounted on a rotating stage which controls the angle of incidence
,
q
i
,
from the
source on
to the sample
. The detector is mounted on a rotating arm to collect the specular reflected
light. Polarizers at the source output and detector input allow polarization
-
dependent
measurements. (
E
)
Scanning electron microscope (
SEM
)
images of the
a
-
Si photonic crystal slab
with no tilt
(top)
,
and
45
°
tilt
(bottom)
. (
F
) Simulation showing the electric field intensity for 50
°
incident radiation at 17.3
μ
m with the measured
a
-
Si
parameters from the SEM images.
13
of
17
Fig. 2:
Drude
reflectance and
dielectric constant of degenerately
-
doped
n
-
InAs
.
(
A
)
Drude
reflectance of the InAs wafer. The transparent black dots show the measured data and the dashed
red
curve
gives the fit for a Drude
-
like optical response with
푛
=
1.5 x 10
18
cm
-
3
, and
Γ
= 4.5
THz. (
B
) Real and imaginary parts of the dielectric function for the isotropic case. The transparent
dots mark the permittivity values directly calculated from the data
,
and the solid blue and red
curves
show the extracted values from fitting the ellipsometry values
Y
and
D
. The chartreuse
strips mark
EN
Z
wavelength regime where we expect to see large nonreciprocal behavior.