ENHANCED AVIONIC SENSING BASED ON
WIGNER’S CUSP ANOMALIES

Joshua Feinberg
University of Haifa
Non-Hermitian QM 2022, University ofTokyo, Hongo Campus, 4 July 2022

Work done in collaboration with

Rodion Kononchuk, Joseph Knee & Tsampikos Kottos

op on Random Matrices, Integrability and Complex Systems

d Hashmona, Judean Hills, Israel
October 03—08, 2018

based on: Science Advances 7(2021) eabg8118

Abstracts1Motivation

Avionic sensing?
(not to worry - this is not an engineering talk)

Airplane navigation systems employ precise accelerometers.
The accelerometer measures the instantaneous acceleration of the airplane and feeds it to the flight computer.
The computer then integrates the equations of motion to find the instantaneous location of the airplane in space.

Other uses of high-sensitivity accelerometers:•navigation•gravity radiometry•earthquake monitoring•platform stabilization for space applications•home security (intruder detection)•airbag deployment in cars
•more ….

Typical accelerometers

https://www.k-makris.gr/accelerometer/

Kostas Makris, a Licensed Aircraft Engineer

Typically, accelerometers use linear sensors: the output is (ultimately an electronic) signal proportional to the instantaneous acceleration.

the sensor detects the displacement of a test mass

output (electric signal) = C x input (acceleration)
sensitivity = d(output)/d(input) = C
fixed sensitivity C
want to make C as large as possible, which is naturally limited

Developing detectors that go beyond linear response might lead to marked enhancement of small-acceleration sensing,
without sacrificing the dynamical range of the sensor

RESEARCH ARTICLE

dynamical range of the sensor = ratio between the maximal and minimal acceleration measurable by the sensor
since the (symmetric n-)integration
5. A. Schliesser, O. Arcizet, G. Rivière, G. Anetsberger, T. J. Keppenberg, Resolved-sideband
cooling and position measurement of a micromechanical oscillator close
efore, the linear polarization of theto the Heisenberg uncertainty limit. Nat. Physics 5, 509–514 (2009).
tion
from the interface.
same
a prominent
exampleAtof thesuchdetectors:
resonantsystems
near
N-th
order
exceptional
point
degeneracy:
6. C. A. Regal, J. D. Teufel, K. W. Lehnert, Measuring nanomechanical motionof the critical angle_
qc of incidencewith a microwave cavity interferometer. Nat. Physics 4, 555–560 (2008).
1
_
~2
~2
where Ap = 2 n √2 (n − 1)−2 (see7. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, W. P. Bowen, Biological
1/N
measurement
beyond the quantum
limit. Nat.
7, 229–233 (2013).
n aEq.
7, we
derive the~ following
small
perturbation
activates sub-linear
response~ Photonics
8. T. Kato, Perturbation Theory for Linear Operators (Springer, 2013).
mponent of the reflected beam<latexit sha1_base64="HDcgX/KD4MHjesjKeV1Uzyc0sZM=">AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBVU2kqMuiG1dSwT6giWUynaRDJzNhZiKU0I0bf8WNC0Xc+g/u/BunbQRtPXDhcM693HtPkDCqtON8WYWFxaXlleJqaW19Y3PL3t5pKpFKTBpYMCHbAVKEUU4ammpG2okkKA4YaQWDy7HfuidSUcFv9TAhfowiTkOKkTZS1973SKIoE/wuc4+vRxB6UQR/tK5ddirOBHCeuDkpgxz1rv3p9QROY8I1Zkipjusk2s+Q1BQzMip5qSIJwgMUkY6hHMVE+dnkixE8NEoPhkKa4hpO1N8TGYqVGsaB6YyR7qtZbyz+53VSHZ77GeVJqgnH00VhyqAWcBwJ7FFJsGZDQxCW1NwKcR9JhLUJrmRCcGdfnifNk4p7WqneVMu1izyOItgDB+AIuOAM1MAVqIMGwOABPIEX8Go9Ws/Wm/U+bS1Y+cwu+APr4xue15gG</latexit>✏⌧1<latexit sha1_base64="E5eLsBbqB3eJOxUxComSejlNM3A=">AAAB9XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE1GPRi8cK9gOaWDbbSbt0swm7G6WU/g8vHhTx6n/x5r9x2+agrQ8GHu/NMDMvTAXXxnW/ncLK6tr6RnGztLW9s7tX3j9o6iRTDBssEYlqh1Sj4BIbhhuB7VQhjUOBrXB4M/Vbj6g0T+S9GaUYxLQvecQZNVZ68DHVXCSS+EIQr1uuuFV3BrJMvJxUIEe9W/7yewnLYpSGCap1x3NTE4ypMpwJnJT8TGNK2ZD2sWOppDHqYDy7ekJOrNIjUaJsSUNm6u+JMY21HsWh7YypGehFbyr+53UyE10FYy7TzKBk80VRJohJyDQC0uMKmREjSyhT3N5K2IAqyowNqmRD8BZfXibNs6p3UT2/O6/UrvM4inAEx3AKHlxCDW6hDg1goOAZXuHNeXJenHfnY95acPKZQ/gD5/MHxBqSCg==</latexit>_

Ap √ine resonance+ m ) d km · dsplittingkn(8)

ns of angle of incidence from theurther_expand thesquare
root
ap_ _
e_
8 as √e + m ≈ √m + 2 m . Substitut√
ogether with the Gaussian form ofat the expression for the reflection
arized wave. The same analysis canyielding the equations_✏

9. M. V. Berry, Physics of non-Hermitian degeneracies. Czechoslovak J. Phys. 54, 1039–1047(2004).
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exceptional points: Application to microcavity sensors for single-particle detection.
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Nat. Photonics 11, 752–762 (2017).
13. M. Parto, Y. G. N. Liu, B. Bahari, M. Khajavikhan, D. N. Christodoulides, Non-Hermitianand topological photonics: Optics at an exceptional point. Nanophotonics 10, 403 (2021).
14. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides,
M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points. Nature 548,
187–191 (2017).

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cooling andpositionmeasurementof a micromechanicaloscillator closedetection.
exceptional
points:Application
to microcavity
sensors
for single-particleanodd
functionofn.
Therefore,thelinearpolarization
of
the
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Y.
G.
N.
Liu,
B.
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ident wave Eq. 6, we arrive at the expression for the reflection
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ase
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and
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Opticsat
anexceptional
Nanophotonics
sientpreserved
thereflection frominterface.
At the cansamefor the during
incident
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angle(ethat≪ atm),thewevicinity
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apof
incidence
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of
the
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q
_
c
implementation
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e for
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_
_ _
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athigher-order
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form
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(
)
(
)
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enhance
4
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_ for
d the
d of
√ 0the0 reflection
13. 8.
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Parto,
G.
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in
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eEq.
for7,the1_km · d_∫ EEp sGlight· =(13_ 0−yielding
Ap √ethe+ them(symmetric)Gdequations
kn n-)integration(8) 17.10.
Ep =that
wes-polarized
find
since
5.M.A.
Schliesser,
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4)
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a micromechanicaloscillator closedetection.
─
·─
·
k
w
·
e
(9)
+
A
Im(
r d ) = Ad of
187–191(2017).
0
odd
functionn.
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of
the
_
d
0
exceptional
Application
tomicrocavity
sensorsforsingle-particle√
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Sagnaceffect.
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9/4 1_
3__
2 √Gpkw
2pG√
totheRev.
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0
(
)
(
)
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4
4
case
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At
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same
_
─
─
− Ad · 9/4 _ √k0w0 · eRe(r d ) = 1 − Ad · 3/4
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pk
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2
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√
lhave
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_
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a microwave
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2 (seee+m≈.
Substitutgwe
parenthesis
of
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as
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−in
min
)the
≈avionic
1 − Athe
e
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_
d
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G
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)
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)
this
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derivethefollowing
4
4
2pn
_
_
─
─
)
=
A
·
·
k
w
·
e
(9)
+
A
Im(
r
0
_
d
d
d
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√
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ident
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2 √pk0w02 √p
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glefor∣e∣
issmalleror equal
to halfofthe Gaussian beamr s-polarized
in than
ne
the
light
yieldingtheequations
=
∫
E
·
(1
−
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e
+
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)
d
k
·
d
k
(8)
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ebasis
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_
─
─
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1with− Adldeviations· being
·
k
w
·
e
−
A
Re(
r=of
15,
014001_d )_d0awavelength,
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)
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aluationof m),
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gyroscopes
enhanced
Sagnacsensitivity.
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e
+
m
≈
m
+
.
Substitutnrized
theparenthesis
of
Eq.
8
as
beam
their
comparison
withtheFresnelle ∣e∣Gaussian
is smaller
thanand
or
equal
to
half
of
the
Gaussian
beam
√
√
12.
L.
Feng,
R.
El-Ganainy,
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photonics
basedonparity-time
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_
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l
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(
)
(
)
tter
expressioninEq.
8
togetherwiththeGaussian
of
4
4
tions
the
associatedlinearapproximation
in
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is re=.
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divergenceesnce:Im(
∣and
er∣≤
ϑ
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pn
w
_
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)
=
A
·
k
w
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1−·─
e(rPSD
30. H. Wen, G. Skolianos, S. Fan, M. Bernier, R. Vallée, M. J. Digonnet, Slow-light fiber Braggl) A_ √k 0w 0 · ed0) = S(fheasd (29,32)_ − Ad · ─our proposal: accelerator with enhanced sensing capability based on sub-linear variations of scattering cross-sections
to small perturbations.
sub-linear response occurs due to Wigner’s cusp anomaly (WCA) of the cross-section at threshold between an open
and closed scattering channel
then use the sub-linear Puiseux expansion for the reflectance/transmittance near threshold for hypersensing

The advantage of our proposal over the exceptional-point based gyros: our WCA protocol is based on intensity
measurements (transmittance/reflectance) and not on resonant shift. The latter is sometimes masked by broadening of
the transmission (or reflection) spectrum, and also in the presence of gain elements, by the unavoidable addition ofquantum noise.
We thus propose WCA-based sensors as simple, yet hypersensitive devices for a variety of applications ranging fromavionics to bio- and chemical sensing.

fundamental principle behind WCA(L&L vol 3)(I shall explain it for NRQM, but this holds also to many phenomena of wave propagation, in particular, of EM waves)pf
d f i = |ff i |d⌦fpi<latexit sha1_base64="UnQJXhhltBTPjJ9ZqO4+sqjNQsw=">AAACHHicbZDLSgMxFIYzXmu9jbp0EyyCqzJTi7oRim7cWcFeoFOHTCZpQ5OZIckIZVrfw42v4saFIm5cCL6N6WWhrQcSPv7/HJLzBwlnSjvOt7WwuLS8sppby69vbG5t2zu7dRWnEpMajnksmwFShLOI1DTTnDQTSZAIOGkEvcuR37gnUrE4utX9hLQF6kSMMoy0kXz7OPQU6wjkZ5QN4Tkc0DEN7koPHpUIZ4lPh+YyZuhdC9JBPvXtglN0xgXnwZ1CAUyr6tufXhjjVJBIY46UarlOotsZkpphToZ5L1UkQbiHOqRlMEKCqHY2Xm4ID40SQhpLcyINx+rviQwJpfoiMJ0C6a6a9Ubif14r1fSsnbEoSTWJ8OQhmnKoYzhKCoZMEqx53wDCkpm/QtxFJhNt8sybENzZleehXiq6J8XyTblQuZjGkQP74AAcARecggq4AlVQAxg8gmfwCt6sJ+vFerc+Jq0L1nRmD/wp6+sHDj+imw==</latexit>

scattering cross section from an initial channel i to final channel f:assume time-reversal invariance T

ff i = fi⇤f ⇤<latexit sha1_base64="qFykGmkymXT7oZaS3TrJUnyuVho=">AAAB+nicbZDLSsNAFIZPvNZ6S3XpZrAI0kVJpKgboejGZQV7gTaUyXTSDp1MwsxEKTGP4saFIm59Ene+jdM2C239YeDjP+dwzvx+zJnSjvNtrayurW9sFraK2zu7e/t26aClokQS2iQRj2THx4pyJmhTM81pJ5YUhz6nbX98M623H6hULBL3ehJTL8RDwQJGsDZW3y4F/TRgGbpCBlglqGR9u+xUnZnQMrg5lCFXo29/9QYRSUIqNOFYqa7rxNpLsdSMcJoVe4miMSZjPKRdgwKHVHnp7PQMnRhngIJImic0mrm/J1IcKjUJfdMZYj1Si7Wp+V+tm+jg0kuZiBNNBZkvChKOdISmOaABk5RoPjGAiWTmVkRGWGKiTVpFE4K7+OVlaJ1V3fNq7a5Wrl/ncRTgCI7hFFy4gDrcQgOaQOARnuEV3qwn68V6tz7mrStWPnMIf2R9/gAhTpNH</latexit>(reciprocity)

inverts signs of all momenta (and spins)pi
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2<latexit sha1_base64="3/sVDbSBsw4+Enorj7ydGvDvq84=">AAACS3icbVBLSwMxGMxWq3V9VT16CRZBeii7UtSLUPTizQr2Ad26ZNOkDU12lyQrlGX/nxcv3vwTXjwo4sH0cahtBwLzzXxDkglizpR2nHcrt7ae39gsbNnbO7t7+8WDw6aKEolJA0c8ku0AKcJZSBqaaU7asSRIBJy0guHt2G89E6lYFD7qUUy6AvVDRhlG2kh+MbA9KhFO3SyNffp0nsHp3PMU6wvkp5RlmZnuBekjn2bwGs4F2IoAK9PyXIRlfrHkVJwJ4DJxZ6QEZqj7xTevF+FEkFBjjpTquE6suymSmmFOMttLFIkRHqI+6RgaIkFUN510kcFTo/QgjaQ5oYYTdT6RIqHUSARmUyA9UIveWFzldRJNr7opC+NEkxBPL6IJhzqC42Jhj0mCNR8ZgrBk5q0QD5CpRpv6bVOCu/jlZdI8r7gXlepDtVS7mdVRAMfgBJwBF1yCGrgDddAAGLyAD/AFvq1X69P6sX6nqzlrljkC/5DL/wGpMbUA</latexit><latexit sha1_base64="JoUWCaRZlDXdIKyojITm2emv4D0=">AAACEXicbVC7TsMwFHV4lvAKMLJYVEhVhyqpKmBBqmBhLIg+pDZEjuu0Vp04sh1QlfYXWPgVFgYQYmVj429I0gzQciRL555zr67vcUNGpTLNb21peWV1bb2woW9ube/sGnv7LckjgUkTc8ZFx0WSMBqQpqKKkU4oCPJdRtru6DL12/dESMqDWzUOie2jQUA9ipFKJMcoTTwn9uh0cleF5zAtaNkrZ2Xvhg6GCgnBH6CuO0bRrJgZ4CKxclIEORqO8dXrcxz5JFCYISm7lhkqO0ZCUczIVO9FkoQIj9CAdBMaIJ9IO84umsLjROlDj4vkBQpm6u+JGPlSjn036fSRGsp5LxX/87qR8s7smAZhpEiAZ4u8iEHFYRoP7FNBsGLjhCAsaPJXiIdIIKySENMQrPmTF0mrWrFOKrXrWrF+kcdRAIfgCJSABU5BHVyBBmgCDB7BM3gFb9qT9qK9ax+z1iUtnzkAf6B9/gBi4ZwW</latexit>

22|ff i | = |fi⇤f ⇤ | )1 d fi
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pf ⇠

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incident flux for threshold for f* —> i*d i⇤f ⇤
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1 d fi
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demonstrates the well-known fact that in a scattering process between propagating modes carrying currents
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the transmittance isInelastic Collisions§147of the second terms contain a non-zero contribution only from the 616
isotropie
part of the amplitude /*(#), the s-wave partial elastic scattering amplitude
Depending on whether the angle 2δο — a is in the first, second, third or
fourth quadrant, the energy dependence of the cross-section described by(e2io
obtainchannel
for the totalelastic scattering cross-section
From
L&LWe
v3,thussinglethreshold0— \)ßikt.
this formula has the forms shown in Fig. 50a, b, c, d. In every case there arenear the threshold
two branches lying on either side of a common vertical tangent.

σ =at-2AV(\E-Et\)

2

f o r £ >Eu
sin Sosin So cos So for E}(147.10)do-This has the form (a) or (b) in Fig. 50, for positive and negative sin So cos So
respectively.
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the particles have spin, of course, the formulae are quantitatively different,IΛ

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WCAs in different physical systems

RESULTS

Fig. 1. Various WCA settings. (A) A WCA describing a square-root behavior of the scattering cross section in the vicinity of a channel threshold. Various forms of WCA are
WCAs in different physical platforms
represented by different line styles (for details, see the Supplementary Materials). (B) Scattering of a quantum particle from a step potential of height U. (C) Near TIR of a
The simplest example leading to WCA is one-dimensional scattering
monochromatic wave from the interface between two dielectric media with reflection indices ni = nH > nf = nL. (D) Two CROW transmission lines with resonant frequency
of a nonrelativistic particle by a step potential of height U extending
detuning U. (E) A waveguide whose (right) portion is exposed to pressure variations leading to a distortion of its cross section, which induces a threshold WCA.

B

A

1 of 8_over x ≥ 0 (see Fig. 1B). For x < 0, the energy dispersion relation is
E = k2i (units of ℏ2/2m = 1) where E and ki are the incident_ energy
and the wave vector, while for x > 0, we have kf = √E − U . Threshold occurs at energy Et = U, signifying a transition from a real (forE > Et) to an imaginary (for E < Et) wave vector kf (corresponding to_an evanescent wave). Above(below) threshold
E > U(E < U), the2kf − ki
E
−U
_
≈
1
−
4
(R =_
1), while the correflectance is R = (_√U
kf + ki)
ki kf
E−U
_
≈
4
(T = 0), which
responding transmittance is T = 4 _2√U(kf + ki)nicely demonstrates the leading square-root term of the systematic
Puiseux expansion of these quantities.
The analogous scenario in optics is associated with light scattering through a dielectric interface from a medium with high refractive index ni = nH to a medium with a low refractive index nf = nL(see Fig. 1C). Assuming that the direction of the normal from theboundary plane is along the z axis, the problem is rotation-invariant
around this axis. Consequently, without loss of generality, the directionof propagation can be taken to lie in the xz plane. The
dispersion____________
relations on each side of the interface are k i,fz = √(_wc ni,f) 2 − (k x) 2 ,where k = (k x, k y, k z). Furthermore, k x = w_c n i sin q i, where qi is theincidence angle with respect to the normal to the interface andx2where Aδ = 2 γδ√2 (n − 1) , n = n /n is a refractive index ratio~2−1_2 ~

HL

between the two media, d = p, s denotes the s- and p-polarized light,
respectively, and gd=s = 1, whileUgd=p =n~ n. These equations showEthat this scattering process results in the formation of a WCA associated with the opening/closing of a scattering channel kzf at a critical incidence angle qc.
WCAs are ubiquitous and can be implemented in a variety ofother photonic platforms. We can, for example, implement them in
one-dimensional coupled resonators optical waveguide (CROW)
arrays at the vicinity of a band edge. A simple realization is shownin Fig. 1D, where two CROW arrays with resonant detuning 0 < U< 2∣t1 + t2∣ and coupling constants t1,2 < 0 are brought together.
When the detuning U varies (e.g., due to exposure of one of the two
CROWs to a biological agent or changes of the ambient temperature orillumination), the transmittance/reflectance exhibits WCA for fre) common trans)("squencies near the band edges associated
with
the
o
c
$"
2
+
+
! =the Supplementary Materials formission frequency domain (seedetailed analysis). Another implementation of WCA involves a uniform
multimode waveguide (see Fig. 1E) in the vicinity of a mode threshold.
In this case, cross-sectional variations (e.g., due to pressure varia-

D

Scatteringcross section

leyaifaMt.
Mid-of quantum noise into the system (11, 17, 18). We therefore propose
the WCA as a simple, yet powerful hypersensing platform for a variety
of applications ranging from avionics to bio- and chemical sensing.

Downloaded from http://advances.sciencemag.org/ on June 4, 2021

ge,
the
een

WCA

Pertu

Fig. 1B). For x < 0, the energy dispersion relation is
2al., Sci. Adv. 2021; 7 : eabg8118 4 June 2021
kf et
ℏ /2m = 1) where E and ki are the incident_ energy
ector, while for x > 0, we have kf = √E − U . Threshatransition
from
a
real
(for
nergy Et = U, signifying
)
s()!
o
c
$
= 2 ! E < Et) wave vector kf (corresponding toaginary! (for_wave). Above(below) threshold
E > U(E < U), the
2
k_
E
−U
f − ki_1), while the cor= (k + k ) ≈ 1 − 4 √ U (R =_
f

i

√

−2 ~
2
~2
Perturbation,!
where Aδ = 2 γδ 2 (n − 1) , n = n H / n L is a refractive index ratio

d2

d1
between the two media, d = p, s denotes
the s- and p-polarized light,~
respectively, and gd=s = 1, while gd=p =n n. These equations showthat this scattering process results in the formation of a WCA asz
sociated with the opening/closing of a scattering channel kf at a critical incidence angle qc.
WCAs are ubiquitous and can be implemented in a variety of

Downloaded from http://advances.sciencemag.org/ on Ju

L
H

Scatteringcross section

2021

expansion of these quantities.
arrays at the vicinity of a band edge. A simple realization is shown=ThenHanalogous
to a medium
with
a
low
refractive
index
n
=
n
When
the
detuningUvaries(e.g.,
duetoexposure
of
one
of
the
two
x niPuiseuxfL
scenario in optics is associated with light scatter- in Fig. 1D, where two CROW arrays with resonant detuning 0 < U
1C).
that thedirectionof the withnormalfrom theto a biologicalchanges
the ambient temperature or+ t2∣ and coupling
constantsagent
t1,2 < 0orare
brought oftogether.
ingAssuming
through a dielectricinterfacefrom a medium
high refrac< 2∣t1 CROWs
C
B
mediuma low refractive
index ninvariant
When thedetuning
U varies(e.g.,
due to exposure of one of the twoexhibits WCA for fretive D
index
ni = ntheEy planeis along
thewith
problemis rotationillumination),thetransmittance/reflectance
H to za axis,
f = nL
(see
Fig.
1C).
Assumingthat
thedirectionofthenormal
from
the
CROWs
to
a
biologicalagentorchangesoftheambient
temperatureor
his axis. Consequently, without loss of generality, the direction quencies near the band edges associated with the common transboundary plane is along the z axis, the problem is rotation-invariant illumination), the transmittance/reflectance exhibits WCA for fregationcanto lie without
in theloss
xzofA
plane.
The
dispersion
missiondomainwith(seeSupplementary
Materials for____________
aroundthisbeaxis.taken
Consequently,
generality,
the direction
quencies
near the frequency
band edges associatedthe thecommontransz
x 2
w
_
2
n
L
n
propagation
canthebe taken
to lie in are
the xzk i,fplane.
The
dispersion
missionfrequency
domain(seetheSupplementary
Materials
for
=
n
−
(
k
)
,
s onofeachside ofinterfacedetailed
analysis).
Another
implementationofWCA
involvesauniform()
H____________
i,f
c
√
z
w
_
2 − (k x) 2 ,
x
y
z
x
w
=
n
_
relations
on
each
side
of
the
interfacearekdetailedmultimodeanalysis). Another
implementationofWCA
involves
a uniformi,f)i,f q√,( cwhere
k
=
n
sin
q
is
the
= (k , k , k ). xFurthermore,
waveguide
(see
Fig.
1E)
in
the
vicinity of a mode threshold
i
i
i
y z
xc w
_
where k = (k , k , k ). Furthermore, k = c n i sin q i, where qi is the multimode waveguide (see Fig. 1E) in the vicinity of a mode threshold.
Ue anglewithrespect
to thenormalinterface
thiscase, cross-sectional(e.g.,
E
incidenceangle with respect
to thenormalto
to thethe interface
and and
In this In
case,cross-sectionalvariations (e.g., variations
due to pressure
varia-due to pressure variax x. Here, threshold
threshold behavior
Snell’s law guarantees
continuity
tions) will
inducetheinduce
closing/openingof a channel at one portionofbehavior
aw guarantees
continuityof kof. kHere,tions)will
theclosing/openingofachannelatone
portionofn_nL ), corresponding to_=q=arcsinoccursatthecriticalangle
q
the waveguide,
which, in turn,will beinsensedas transmittancechangesic(nt thez critical angle q i = qc = arcsin(nH), corresponding to
the waveguide,
which,
turn,will be sensed
as transmittance changes
kf = 0. The reflection r and transmission t amplitudes are given by following a sublinear WCA (see the Supplementary Materials).
Thethereflectionr andastransmission
t amplitudes
given
following a sublinear WCA (see the Supplementary Materials).
kf bysame expressions
in the quantum step
example, withare
ki andz
the criticalq = qc − ewith(0 < ek≪
substituted by
implementation of a WCA avionic sensor
and Experimental
kf
expressions
askin
thebelowquantum
stepangle
example,i 1),
i,f. Justz expand r, t in a Puiseux series, leading to the polarization- As proof of concept, we have demonstrated the hypersensitive nawecan≪ 1), Experimental implementation of a WCA avionic sensor
ed by ki,f. Just below the critical angle q = qc − e (0 < eWCA
A settings.
(A) Areflectance
WCA describing
a square-root behavior of the scattering crossthe vicinityof a channelmeasurements.
threshold. Various
forms of WCA are
dependent(transmittance)turesectionof WCAinsensors
for acceleration
The experimenxpand
t in(fora details,
Puiseux
leading to
the polarizationAs proof
of concept,
wehavedemonstratedthehypersensitive
natal
platform(see
Materials
andMethods)
is
shown
in
Fig.
2
(A
and
C)
erent
liner,
stylessee series,
the Supplementary
Materials).
(B)
Scatteringofaquantum
particle
from
a
step
potentialofheight
U.
(C)
Near
TIR
of
a
_
_
E
D
2
2
2 Adielectrict∣ ∝ with
2 Ad √reflectione(2) nandR(transmittance)≡ ∣betweenr∣ ∝ 1 −twont from
reflectanceture
WCA
sensors
forbehavior
acceleration
measurements.
square-root
thresholdduewithto totalinternald √e , T ≡ ∣ave
the interface
media
indicesnfthe= nof
Two CROW
transmission
lines
resonant
frequency The experimeni = nuses
H>
L. (D)(TIR).
It involves
a whichGaussianlinearpolarized
beam,
aveguide whose (right) portion is exposed to pressure variations leading toreflection
a distortionof its crosssection,inducesaMethods)
thresholdWCA.
talplatform(see
Materials
and
is
shown
in
Fig.
2
(A
and
C)
_
_
2
2
− 22021;
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due
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total
interna(,)T") ≡ ∣4s
Kononchuk
2021
2
of
8
!=++
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reflection(TIR).
It
involvesaGaussianlinearpolarizedbeam
1_

2 of 8

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_

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Scatteringcross section

light scatterscatterlighthigh refracrefrachh high
index
=n
nL
ndex nnff =
L
mal
from
the
mal from theioninvariant
on-invariant, thethe direction
directionhedispersion___________
he dispersion___________
x 2
2
n
−
(
k
x) 2 ,)i,f2ni,f) − (k ) ,here qi is thehere qi is thenterface andnterface andold behavior
old behavior
responding toesponding tos are given by
are given bywith ki and kf
with ki and kf
e (0 < e ≪ 1),
e (0 < e ≪ 1),
polarizationpolarization-

zat a critsociatedwiththeopening/closingofascatteringchannelksociated with the opening/closing of a scattering channel kf f at a critical
ical incidence
incidence angle
angle qqcc..
WCAs
WCAs areare ubiquitousubiquitous andand cancan bebeimplemented
implementedininaavariety
varietyofofotherB platforms.
other photonic
photonic
platforms.We
Wecan,
can,forforexample,
example,implement
implementthem
theminin
one-dimensional
one-dimensional coupledcoupled resonators
resonatorsoptical
opticalwaveguidewaveguide(CROW)(CROW)
arraysrealization
isisshownarrays at
at thethe vicinity
vicinity ofof aa bandbandedge.
edge.AAsimple
simplerealization
shown
A
in
in Fig.
Fig. 1D,
1D, where
where two
two CROW
CROWarrays
arrayswith
withresonant
resonantdetuning
detuning00<<UU<
and coupling
coupling constants
constantst1,2
t1,2<<00arearebrought
broughttogether.
together.
< 2∣t2∣t11 ++ tt22∣∣ andWhenUvaries(e.g.,
duetoexposure
of
one
of
the
two
When thethe detuning
detuningUvaries(e.g.,
duetoexposure
of
one
of
the
two
U
E
CROWs
to
a
biologicalagentorchangesoftheambient
temperature
or
CROWs to a biological agent or changes of the ambient temperature orillumination),thetransmittance/reflectance
exhibitsWCA
forfreillumination), the transmittance/reflectance exhibits WCA for frequencies
quencies near
near thethe bandband edges
edges associatedassociatedwith
withthethecommon
commontranstransmission
frequency
domain(seetheSupplementary
Materials
formission frequency domain (see the Supplementary Materials fordetailed
analysis).
Another
implementationofWCA
involvesauniform
detailed analysis). Another implementation of WCA involves aWCA
uniform
multimode waveguide (see Fig. 1E) in the vicinity of a mode threshold.
multimode waveguide (see Fig. 1E) in the vicinity of a mode threshold.
In this case,Dcross-sectional variations (e.g., due to pressure variaIn this case, cross-sectional variations (e.g., due to pressure variations) will induce the closing/opening of a channel at one portion oftions) will induce the closing/opening of a channel at one portion of
the waveguide, which, in turn, will besensedastransmittancechanges))(
the waveguide, which, in turn,will$"cosbe" sensed as transmittance changes
2++! = (see the Supplementary Materials).
following a sublinear WCA
following a sublinear WCA (see the Supplementary Materials).

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Experimental implementation of a WCA avionic sensor
Experimental implementation of a WCA avionic sensor
As proof of concept, we have demonstrated the hypersensitive naAs
proof
of
concept,
wehavedemonstratedthehypersensitive
nature of WCA sensors for)acceleration measurements. The experimen()! acceleration measurements. The experimens
ture
of
WCA
sensorsforoc
2$!
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talplatform(see
Materials
andMethods)
is
shown
in
Fig.
2
(A
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C)
and uses the square-root threshold behavior due to total internal
and
uses
the
square-root
thresholdbehavior
due
to
total
internal
reflection (TIR). It involves a Gaussian linear polarized beam,reflection (TIR). It involves a Gaussian linear polarized beam,

)
)
(
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s
o
c
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SCIENCE ADVANCES | RESEARCH ARTICLE

C

B

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E
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d2

Fig. 1. Various WCA settings. (A) A WCA describing a square-root behavior of the scattering cross section in the vicinity of a channel threshold. Various forms of WCA are
represented by different line styles (for details, see the Supplementary Materials). (B) Scattering of a quantum particle from a step potential of height U. (C) Near TIR of a
monochromatic wave from the interface between two dielectric media with reflection indices ni = nH > nf = nL. (D) Two CROW transmission lines with resonant frequency
detuning U. (E) A waveguide whose (right) portion is exposed to pressure variations leading to a distortion of its cross section, which induces a threshold WCA.

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SCIENCE ADVANCES | RESEARCH ARTICLE

ACPolarizing
beam
splitterPowermeter 2!/4 plate

Prism 2

NonPolarizing
50:50 beam
splitter

Laser

Added
mass

Polarizing
beam!/4 platesplitterPrism 2

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B

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Differential signal, IPowermeter 1

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mirrorwithspringmassPowermeter 1

b

0.4
0.3
0.2
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0.1

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Puiseux expansion

0

0.02

0.04

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0.06

(g)

0.08

a

0.4
0.3
0.2
0.1

0

data1
data3

0.02

0.04

In-plane Acceleration,

Fig. 2. WCA accelerometer. (A) Experimental implementation of the TIR as a WCA protocol for a hypersensitive accelerometer. (B) Magnification of the mirror with theattached test mass that is coupled to two torsional springs. (C) Schematic description of the platform shown in (A). (D) Measured differential signal I versus in-plane accelWTheeration a (filled blue circles).
system is calibrated to demonstrate a TIR when it is at rest. Any acceleration will lead to a WCA. The black line indicates the Puiseux
1
expansion near WCA. Photo credit: (A and B) Rodion Kononchuk, Wesleyan University.

A
Experiment
A DGaussian
V A N Cbeam
ES |

10

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B

10 2

0

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1

10

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10

-2

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10

1

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-3

10

-4

-3

10-3
10

10

-4

10

10

-3

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10

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10

In-plane acceleration,

-1
10

-1

(g)

10
10

(g)

10

1

10

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7.5 mradExperiment
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0.3 mrad0

1.5 mrad-1/2

7.5 mrad
0

10

10 -1

0

0

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A
10

0

1.5 mrad

1/2

10 -1

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0.3 mradRESEARCH ARTICLE

1

Differential signal, I

Differential signal, I

SCIENCE

10

10 2

0

Sensitivity,

10

B

-1

10

-4

10 -4
10

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-2

10 -3
10

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10

(g)

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In-plane acceleration,

(g)

10 0

wavelength and/or a larger waist w0 will further reduce the GHS

the ambient temperature, m = 0.027 kg is the mass, wn is the natural

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Fig. 3. Beam divergence effects and WCA-sensitivity. (A) Measured differential signal I versus in-plane accelerations a (blue filled circles) reported in log-log plot. The
0 1
10
theoretical calculation that incorporates the Gaussian beam divergence of 0.3 mrad is shown in the red solid line. The
0 black dotted line has a slope of /2, while the black
0.5
dashed line indicates the linear slope of the conventional acceleration sensor with the same maximum
output 10
and with linear sensitivity. The green double-sided arrow
indicates the dynamic range enhancement of the proposed WCA sensor. (B) Measured sensitivity of the accelerometer c ≡ dI/da (blue filled circles). The red solid line
data1
describes the theoretical curve that takes into consideration the Gaussian beam divergence associated with the characteristics of our lasing source (0.3-mrad beam diverdata2
gence). In the same figure, we also report with a dark red (purple) dashed-dotted line the theoreticallycalculated sensitivity for a lasing source with a -4
beam
-3 divergence-2 -1 0of
10
10
10
10 10
1
−1
-4
-3
-2
-1
0
1.5 mrad (7.5 mrad). The black dotted line has a slope of − /2, while the black dashed line indicates
c =10
0.6710g oflinear accelerometer10 10
10a conventional
data3 a constant sensitivity
In-plane Acceleration,(see the main text). The green double-sided arrow indicates the sensitivity enhancement of the proposed WCA
sensor.Acceleration,
In-plane(g)(g)

Figure S1: Reflection of Gaussian beam from a dielectric interface: (A) Real part of thereflection amplitude of the p-polarized Re(rp ) light calculated using Fresnel’s equations (bluedotted line) and taking into consideration divergence effects due to a Gaussian beam shape (redline). (B) Magnification of the domain around the black square frame shown in (A). The blue
dashed line is the linear approximation calculated from Eq. (9) (see main text), while the verticalgreen dashed lines indicate the Gaussian laser beam divergence used in the calculations. (C)The imaginary part of the reflection amplitude of the p-polarized Im(rp ) light calculated using
Fresnel’s equations (blue dashed line) and taking into consideration the divergence effects of
the Gaussian beam (red line). (D) Magnification of the domain around the black square frame
shown in (C). The blue dashed line is the linear approximation calculated from Eq (9) (see maintext), while the vertical green dashed lines indicate the Gaussian laser beam divergence used in
the calculations.

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indicates the region where the bands overlap forming a transmission band of the interface. The
black dashed box indicates the system described by the Hamiltonian H0 .

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T

T

1/2

T

T

0.3

Transmission, T

T

0.3

-3

1.95 100.05
1.95

Frequency,
Frequency,

0

-2

3.9

-0.05
3.9
2 0.1

Figure S4: Transmission Spectrum and WCA for a CROW array. (A) Transmission spectrum of the CROW array interface shown in Fig. S3 (see also Fig. 1D). The parameters used
are t1 = 1, t2 = 2, = 1 and U = 4. The transmission band in this case is within thefrequency range of ! 2 [0, 2]. (B) Magnification of the transmission spectrum at the vicinity of(1)the lower frequency threshold corresponding to !t = 0. In the inset we show in a log-log plot(1)
the dependence of transmittance T for small frequency detunings ✏ = ! !t . (C) Dependenceof the transmittance from the CROW detuning parameter U near its threshold value Ut = 4 for(1)
an incident wave at fixed frequency ! = !t . In the inset we show the transmittance versus✏ = Ut U in log-log plot. (D) Transmission spectrum near the upper frequency threshold(2)(2)!t = 2. In the inset we plot the transmittance versus detuning
✏
=
!
!
in
a
log-log
plot.
tp
The black dashed lines in subfigures (B-D) is proportional to ✏ and is drawn in order to guidethe eye.

T

B

T

Transmission, T

A

A
C 0.6

3.95

3.95

1.9

4

Frequency detuning, U

-0.05
1.9

1.95

1.95

Frequency,

2

A

B

A
1
2

B
1

1
2

n

3

3

2

n

n-1

1
2

n

3

n-1

n

3

n-1

Figure S5: General setting of a multimoded system with WCA. An example of a multimoded
scattering setting where n 1 channels are open while the n-th channel undergoes a transition
from (A) closed to (B) open as an external stimuli enforces variations of the index of refraction
which lead to modification of its threshold.

S3

Cross-Section Cusps Associated with Energy Thresholds

n-1principle of detailed balance, one has the following relation betweenof applications ranging from avionics to bio- an

1

Wave Transport in Complex Systems Laboratory, Department of Physics, Wesley2
For more(physical,
technical& mathematical)
see
our
paper
an University, Middletown,
CTdetails
06459, USA.
Departmentof Mathematics
and Haifa
RESULTS
Research Center for Theoretical Physics and Astrophysics, University of Haifa, Mt.
WCAs in different physical platforms
3
Carmel, Haifa 31905, Israel. Department of Chemistry, Wesleyan University, MidThe simplest example leading to WCA is one-di
dletown, CT 06459, USA.
of a nonrelativistic particle by a step potential o*Corresponding author. Email: tkottos@wesleyan.edu.
Kononchuk et al., Sci. Adv. 2021; 7 : eabg8118

4 June 2021

SCIENCE ADVANCES | RESEARCH ARTICLEAPPLIED SCIENCES AND ENGINEERING

Enhanced avionic sensing based on
Wigner’s cusp anomalies
Rodion Kononchuk1, Joshua Feinberg2, Joseph Knee3, Tsampikos Kottos1*
Typical sensors detect small perturbations by measuring their effects on a physical observable, using a linear response principle (LRP). It turns out that once LRP is abandoned, new opportunities emerge. A prominent example
is resonant systems operating near Nth-order exceptional point degeneracies (EPDs) where a small perturbationNe ≫ e in resonant splitting. Here, we propose an alternativee ≪ 1 activates an inherent sublinear response ∼ √─sublinear optomechanical sensing scheme that is rooted in Wigner’s cusp anomalies (WCAs), first discussed in theframework of nuclear reactions: a frequency-dependent square-root singularity of the differential scattering crosssection around the energy threshold of a newly opened channel, which we use to amplify small perturbations.
WCA hypersensitivity can be applied in a variety of sensing applications, besides optomechanical accelerometry
discussed in this paper. Our WCA platforms are compact, do not require a judicious arrangement of active elements (unlike EPD platforms), and, if chosen, can be cavity free.

ds

ds

1 _fi
1 _
i*f*
_
=
the reaction cross sections _2
2v dWv d Wi* , where the subindexesffi

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