Sub-wavelength Position Sensing Using
Nonlinear Feedback and Wave Chaos
1/30
Seth D. Cohen, Hugo L. D. de S. Cavalcante, and Daniel J. Gauthier
Duke University, Department of Physics, Durham, NC, 27708 USA
MURI
Cohen, et al. Phys. Rev. Lett. 107, 254103 (2011)
2/30
Can we exploit nonlinear dynamics for sensing?
Hypothesis: The sensitivity of nonlinear dynamics and chaos can
give advantages over traditional sensor systems.
Traditional sensor: Microscope
object
light
lens
limiting resolution:
diffraction λ/4
Resolution for visible light ~ 200 nm
Super-resolution = better resolution than diffraction limit
3/30
Our approach to sub-wavelength position sensing
y
input: Vin(t)
wave-chaotic
cavity
RX
nonlinear
electronic
circuit
x
scatterer
TX
output: Vout(t)
measure single scalar voltage: Vout(t)
frequency content of Vout(t) = unique fingerprint of object position (x,y)
1D Resolution: λ/10,000 , 2D resolution: λ/300
4/30
Time-delayed nonlinear feedback
5/30
Time-delayed nonlinear feedback
input = g x(t - τ)
g
nonlinear
operator: fNL
Time-Delay: τ
output = x(t)
Dynamics
described by:
.
x(t) = fNL(x(t), x(t - τ))
Depending on fNL and τ, such systems can self oscillate for g ≥ 1:
6/30
Experimental nonlinear circuit
input = g v(t - τ)
nonlinear
circuit: fNL
g
coaxial cable: τ
Vout
Vin
output = v(t)
Dynamics
described by:
.
v(t) = fNL(v(t), v(t - τ))
nonlinearity
Vout
nonlinear circuit
Vin
Illing, L. & Gauthier, D. J. Ultra-high-frequency chaos in a time-delayed electronic device with band-limited
feedback. Chaos 16, 033119 (2006).
7/30
Nonlinear circuit oscillations
For a given set of system parameters g, τ :
periodic: f
7/30
Nonlinear circuit oscillations
Tune dynamical state by changing system parameters:
τ
periodic: f
τ = τ +δτ
periodic:
f’ = f +δf
8/30
Nonlinear circuit oscillations
Tune dynamical state by changing system parameters:
g
periodic: f1
g
quasi-periodic:
f1 and f2
8/30
Nonlinear circuit oscillations
Tune dynamical state by changing system parameters:
g
periodic: f1
g
quasi-periodic:
f1 and f2
δτ or δg cause shifts in multiple frequencies
We will use quasi-periodicity for sensing
9/30
FM radio:
88 MHz 108 MHz
~ 3.1 m
microwave
:
2.45 GHz
~ 12.2 cm
- simultaneous multiple frequencies in broad range = multiple λ-scales
- minimum λ = 17 cm, diffraction: λ/4 = 4.25 cm
10/30
Ray/Wave Chaos
Ray chaos
Bunimovich stadium: 2D Billiard
particles
rays = trajectories
Rays are exponentially sensitive to initial conditions
+
Non-repeating trajectories
+
Deterministic
Ray Chaos
11/30
Position - Momentum
chaotic billiard
non-chaotic billiard
Generic cavities tend to display ray chaos.
Stone, A. D., Physics Today (American Institute of Physics) August 2005 37-43.
12/30
Wave chaos: water tank
Liquid vibrated from the sides, shine light, image standing waves:
chaotic tank
Waves form complex
interference patterns.
non-chaotic tank
Waves form regular
interference patterns.
Hans-Jürgen Stöckmann, Quantum Chaos: an introduction, Cambridge University Press, 1999.
13/30
Wave chaos: EM simulations
14/30
Inject continuous-wave EM periodic (monochromatic) signal into cavity:
asymmetric:
chaotic
interference
pattern
symmetric:
regular
interference
pattern
Stone, A. D., Physics Today (American Institute of Physics) August 2005, Cover.
15/30
Experiment: ¼ stadium
15/30
Characterization: pulse response
15/30
Delay Distribution
τ1, τ2, τ3,...
g1, g2, g3,...
Complex
response is
indicative of
wave chaos.
(τi, gi)
(τ1, τ2, τ3,..., g1, g2, g3,...) ~ delay distribution
16/30
Sub-wavelength position sensing:
wave chaos + nonlinear feedback
17/30
Wave chaos, nonlinear feedback
chaotic
cavity
τ1, τ2, τ3,...
g1, g2, g3,...
RX
TX
nonlinear
feedback loop
Vout = v(t)
coaxial cable: τ
g
nonlinear
circuit
Vin = g v(t - τ)
18/30
Nonlinear feedback + wave chaos
nonlinear
feedback
system with a
distribution of
delays
Vout = v(t)
τ1, τ2, τ3,...
g1, g2, g3,...
RX
TX
g
nonlinear
circuit
Vin = Σ giv(t - τi)
i
Position sensing system
τi(x,y)
gi(x,y)
n ≈ 10
Vout
g
nonlinear
circuit
filter Vout:
20 MHz - 2 GHz
λmin = 15 cm
Recall: changes in τi and gi cause frequency shifts in Vout
Goal: quantitatively track (x,y) using frequency shifts
19/30
20/30
Why quasi-periodic dynamics?
vout
Goal: quantitatively track (x,y) using frequency shifts:
●
●
f1 and f2 are incommensurate: irrationally related
these frequencies give independent information in x and y
21/30
Why quasi-periodic dynamics?
f1 :
+
f2 :
Goal: quantitatively track (x,y) using frequency shifts:
●
simultaneous irrational frequencies = less blind spots (nodes)
22/30
Photographed experimental setup:
quarter stadium cavity
x
y
water scatterer
1 oz bottle
23/30
Photographed experimental setup:
sealed cavity
x
y
Inside Cavity
24/30
Photographed experimental setup:
nonlinear feedback
transmitted
signal
received
signal
nonlinear
circuit
Vout
TX antenna
inside cavity
RX antenna
inside cavity
25/30
1D and 2D position sensing:
Experimental Results
29
26/30
1D position sensing: x
x1 = object starting position, x2 = x1 + δx
δx
(Δf1, Δf2)
27/30
1D position sensing: x
map: a1 Δf1(x) + a2 Δf2(x) = c0 + c1 x + c2 x2
27/30
1D position sensing: x
map: a1 Δf1(x) + a2 Δf2(x) = c0 + c1 x + c2 x2
RMS position error: 9.2 μm, ~ λ/10,000
27/30
1D position sensing: y
map: b1 Δf1(y) + b2 Δf2(y) = d0 + d1 y + d2 y2
RMS position error: 23.7 μm, ~ λ/10,000
28/30
2D position sensing: x, y
(x1, y1) = object starting position, (x2, y2) = (x1, y1) + (δx, δy)
(δx, δy)
(Δf1, Δf2)
must be
independent:
QP
29/30
2D position sensing: x, y
map using planar fits:
Δf1(x,y) = α1 x + β1 y + ε1,
Δf2(x,y) = α2 x + β2 y + ε2
planar fits
are linearly
independent
29/30
2D position sensing: x, y
map using planar fits:
Δf1(x,y) = α1 x + β1 y + ε1,
Δf2(x,y) = α2 x + β2 y + ε2
planar fits
are linearly
independent
xRMS = 370 μm,
yRMS = 650 μm,
~ λ/300

Our approach:




Summary
30/30
uses the sensitivity of nonlinear feedback and wave chaos
measures multiple spatial degrees of freedom on a
sub-wavelength scale using a scalar, quasi-periodic signal
presents a new alternative to the short list of superresolution techniques.
Future work:



3-D position sensing, all-optical system
sensing with multiple objects,
temperature, pressure sensing, …
explore alternative layouts:

room sized cavity?

dynamical chaos?
37
Special Thanks
●
Hugo Cavalcante and Daniel Gauthier
●
Office of Naval Research: MURI #N000014-07-0734
“Exploiting Nonlinear Dynamics for Novel Sensor Networks”
Questions?
Seth Cohen, e-mail: sdc18@phy.duke.edu
Cohen, et al. Phys. Rev. Lett. 107, 254103 (2011).
More complex quasi-periodic signal
2-D: (Δf1, Δf2)
3-D: (Δf1, Δf2, Δf3)
●
(x, y)
(x, y, z)
more frequencies, improve
SNR, better resolution
Other degrees-of freedom:
●
sensitivity to object shape and orientation
●
EM properties of materials
Extensions to optical frequency domain
time-delayed optical feedback
optical ray chaos
Gensty, T. et al., Phys. Rev. Lett. 94,
233901 (2005).
Murakami, A. et al., IEEE J. Quant. Elec. 34,
10 (1998).
Extensions to optical frequency domain
All-optical system using chaotic optical microcavity (COM):
bandwidth expanded to optical frequencies
●
wavelengths scaled to ~102 nm
● potential sensitivity < 1 nm
●