© 2016 The Aerospace Corporation
Using optical speckle in multimode
waveguides for compressive sensing
George C. Valley, George A. Sefler, T. Justin Shaw, Andrew Stapleton
The Aerospace Corporation, Los Angeles CA
3 June 2016
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Outline
• Motivation for Compressive Sensing for GHz-band RF signals
• Compressive sensing
– Sparsity
– Mixing down in dimension
– Recovery
• Electronic CS system
• Photonic CS systems
– Measurement matrix
– Calibration
– Path to photonic integrated circuit for CS
• Photonic CS using speckle in multimode waveguide
• Other speckle-based photonic systems
• Conclusions
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Motivation for compressive sensing
• Nyquist-rate ADCs for GHz-band RF signals …
– generate a tremendous amount of data
• Rapidly fill storage buffers
• Overwhelm processors
• Swamp communication links
– have limited performance
– consume significant power
• Signals of interest in GHz band are often sparse
Frequency domain Frequency domain (tones) Time domain (pulses)
Frequency, GHz Frequency, GHz Time, ps
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Compressive Sensing--pulses
• N is # measurements needed to measure the signal at Nyquist rate
• K is the sparsity (# of pulses, sinusoids,…)
• M is the # CS measurements needed to recover the signal s
N >> K and M > K
• CS theorems show that for certain classes of Measurement Matrix A, one can recover s with high
probability if
M > c K log(N/K) (c ~ O[1])
=
N
Measurement
Matrix
AM measurements
Measurement
vector
y
Sparse signal
vector
s
K non-zero elements
Accurate knowledge of the Measurement Matrix A is critical for CS recovery calculations
y = As
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Compressive Sensing—arbitrary waveforms
• N is # measurements needed to measure the signal at Nyquist rate
• K is the sparsity (# of pulses, sinusoids,…)
• M is the # CS measurements needed to recover the signal s
N >> K and M > K
• CS theorems show that for certain classes of Measurement Matrix A, one can recover s with high
probability if
M > c K log(N/K) (c ~ O[1])
K non-zero elements
Accurate knowledge of the Measurement Matrix A is critical for CS recovery calculations
Inverse
Transform
Matrix
Y-1
=
N
=
Mixing
Matrix AM
Measurement
Vector
y
Sparse Vector
s
y = Ax = A (Y-1s) = Qs s = Yx
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The 3 fundamental aspects of CS
• Sparse signals/images
– Subtract off known background
– Threshold noise
– Transform to a basis in which signal is sparse
• Analog Mixing
– Wideband converter—mix with pseudo-random waveforms (e.g. PRBS)
– Single pixel camera—mix with pseudo-random images
• Signal/Image Recovery
– Need accurate knowledge of analog measurement matrix
– Exploit sparsity to limit solution space
– Wide range of algorithms and codes now available
• Penalized l1-norm
• Orthogonal matching pursuit
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• A sparse vector/matrix–mostly 0’s
• What percentage is sparse?
• Noise—threshold signal
• Sine waves—Transform signal
• Sin( p t)
• Beware of off-grid frequencies
• Sin(1.1 p t)
• Subtract off non-sparse background
Sparsity
There is always a transform that makes a signal sparse—but you may not know it!
DFT
DFT
Signal + Background Background DFT(Signal+Background) DFT(Signal)
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Electronic CS system for GHz-band RF Signals
• Split signal into M copies
• Multiply each copy by different pseudo-random bit sequence (PRBS) of length N
• Integrate for duration of PRBS, sample, and digitize
• Issues:
– Requires M electronic pattern generators
– Size, weight, and power
– Random noise and jitter within PRBS limit recovery
Mishali and Eldar, IEEE Journal of Selected Topics in Signal Processing, Vol. 4, pp. 375 (2010).
Splitter
/Divider
0
1
RF input
0
1
Copy 1
Copy M
yA
Pattern
Generator #1
X
X
Pattern
Generator #M
ADC #1
ADC #M
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Photonic undersampling and compressive sensing
Origins
Before 2008
• Photonic ADC work
• Photonic down-conversion/down-sampling
2008
• Moshe Horowitz’ group Technion: Multi-rate asynchronous sampling
• Johns Hopkins U. Appl. Phys. Lab: Non-uniform sampling
2010 Photonic CS
• Technion group use CS techniques to recover signals
• JH APL group recognizes non-uniform sampling is a form of CS
• Aerospace group proposes parallel multi-rate sampling CS
2011-2016
Approximately 30 papers on Photonic CS
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Serial CS system using PRBS and EOMs
Laser EOM 1 EOM 2 PD ADC
RF signal PRBS
Nichols and Bucholtz 2011
Chi et al. 2012
Yan et al. 2012
McKenna et al. 2013
Chen et al. 2013
Yin et al. 2013
INT
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Parallel CS system using PRBSs and EOM
Still has amplitude and timing jitter of electronics
Proposed WDM pair to obtain simultaneous measurement of all elements in
measurement vector y (Nan et al. 2011)
Pseudo-random bit sequences impressed on cw diode lasers prior to RF signal
PRBSs
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Proposed Parallel CS system using PRBS and
EOM
Still has amplitude and timing jitter of electronics
Proposed WDM pair to obtain simultaneous measurement of all elements in
measurement vector y (Nan et al. 2011)
Pseudo-random bit sequences impressed on cw diode lasers prior to RF signal
yA
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CW or pulsed laser?
Low-jitter pulsed laser removes effect of PRBS jitter
CW laser
Advantages: Simple, efficient, low average power
Disadvantage: Timing jitter of PRBS mapped onto optical intensity
Pulsed laser
Advantage: Timing jitter of PRBS removed by low jitter mode-locked laser
Disadvantages: high peak power on photodiode
Neither system avoids amplitude noise of PRBS generator
Low jitter PRBS
High jitter PRBS
Laser
pulses
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Time stretching/compression to decrease effective
amplitude and timing jitter in PRBS
Time-stretching allows use of lower rate PRBS with less jitter
Demonstrated use of stretching/compression to increase effective rate of PRBS
(Bosworth and Foster Optics Letters 2013)
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Parallel CS system with PRBS addressing 2D
Spatial Light Modulator
MLL
Chirped
FBG
MZM ADCs
Ph
oto
dio
de
Arr
ay
Broadband
Optical Pulse
Modulated
Chirped Optical Pulse
Spatial Light
ModulatorDiffraction
Grating
x
z
y
x
ADC
ADC
ADC
ADC
ADC
ADC
ADC
ADC
Spherical
Collimator Cylin
dri
ca
l L
en
s
Cylindrical
Lens
Row
RF Signal
Wav
elen
gth
Time Row
Colu
mn
Mode-locked
Laser
A y
Valley and Sefler 2010
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Parallel CS system with PRBS addressing 2D
Spatial Light Modulator
MLL
Chirped
FBG
MZM ADCs
Ph
oto
dio
de
Arr
ay
Broadband
Optical Pulse
Modulated
Chirped Optical Pulse
Spatial Light
ModulatorDiffraction
Grating
x
z
y
x
ADC
ADC
ADC
ADC
ADC
ADC
ADC
ADC
Spherical
Collimator Cylin
dri
ca
l L
en
s
Cylindrical
Lens
Row
RF Signal
Wav
elen
gth
Time Row
Colu
mn
Mode-locked
Laser
A y
x
Diffraction Grating
Wavelength-Space mapping
Chirped Pulsed Laser
Time-Wavelength mapping
Detector array integrates RF-PRBS
products over a pulse period
Sampled at pulse repetition rate
High-speed MZM modulates RF
signal onto chirped laser pulses
Mixing Matrix A realized with 2D spatial light mask (SLM)
Each SLM row mixes a different PRBS with RF signalValley and Sefler 2010
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Serial Experimental Demonstration with 1D SLM
• Used 1D liquid-crystal SLM
• RF signals synchronized to laser pulse repetition rate
• Measurements y made sequentially by stepping SLM through rows of A
• RF tones (sparse in frequency domain) and pulses (sparse in time domain) recovered using penalized l1-norm
Valley, Sefler, and Shaw, 2012
Spatial Light
Modulator
Diffraction
Grating
MLL
DCF
MZM
Large-
Area
PDADC
Broadband
Optical Pulse
Modulated
Chirped Optical Pulse
Spatial Light
Modulator
Diffraction
Grating
Spherical
Collimator
Cylindrical
Lens
RF Signal
Mode-locked
Laser
Light
Path
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Parallel CS system using WDMs (AWGs)
Arrayed Waveguide Gratings (AWGs)
and attenuators (ATTNs) form measurement matrix
Matrix of in-line attenuators
PD#M
ADC#M
PD#1
ADC#1
MLL
ChirpedFBG
MZM
BroadbandOptical Pulse
ModulatedChirped Optical Pulse
ATTN
ATTN
ATTN
ATTN
ATTN
ATTN
1xNAWG
Nx1AWG
#1
Nx1AWG#M
1xMSplitter
RF Signal
Mode-locked
Laser
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Serial experimental demonstration using AWGs
• Single row of Mixing Matrix A
• AWG: 96 wavelength channels @ 0.4-nm channel spacing‒ Selected channels blocked to form PRBS
• Delay-line interferometer (DLI) Up-converter
• RF bandwidths from 4 to 20 GHz
Nanoseconds
Op
tica
l In
ten
sity
0 4 8 12 16 20 24 28
MLL MZM
RF Signal
DCF
EDFA PD ADC
I
II
I
I
AW
G
AW
G
Baseband
PRBS
DLI
PRBS
Up-Converter
Low-pass
Integrator
Clock Signal
Broadband
Optical Pulse
RF-Modulated
Chirped Optical PulsePRBS-Modulated
Optical Pulse
Integrated
Pulse
Mode-locked
Laser
AWGs
PD
DCF
MZM
EDFA
Non-uniform sampling
pattern modulates the
stretched laser pulse
and RF signal
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Periodic non-uniform sampling with integration
• Periodic non-uniform sampling and integration
– Single PRBS repeated
• Measurement Matrix has generalized block diagonal structure
• Arbitrary number of measurements yi
• Arbitrary length of Signal x
• Effective sampling rate is PRBS/laser rep rate
• No RF-to-laser synchronization required
• Useful for long duration RF signals (e.g. chirped pulses
.
.
.
=.
A
yx
Generalized block diagonal matrix
w/ PRBS along diagonal
PRBS
PRBS
PRBS
PRBS
.
.
.
.
.
.
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Experimental results for RF chirped pulses
Carrier Recovery vs. Number of Measurements
WDM mixing
• RF Parameters:
‒ Chirp = 20 MHz / 30 ms
‒ Carrier Freq = 2.453 GHz
• PRBS Rep Rate (effective sample rate) = 35 MHz
• Maximum likelihood recovery technique
20 Measurements
272 Measurements80 Measurements
40 Measurements
Location of peak
gives the
frequency
0 200 400 600 800 1000 1200 1400-100
-80
-60
-40
-20
0
20
40
60
80
100
Coset Sample
Spulse 20MHz 3Vpp
yDue to beating of chirp
and sampling grid
Time/Measurement number
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Desirable Properties for Photonic CS System
• Components integrable in one or more photonic integrated circuits
‒ Avoid free-space optics
‒ Minimize fiber-coupled devices
• Static or low-error PRBS generation that can be calibrated
‒ Spatial light modulators
‒ Pulse or bandwidth compression (Bosworth and Foster 2013)
‒ Optical pulse in the center of each PRBS bit (Chi et al. 2012)
‒ WDMs
• Operation as real-time digitizer
‒ Unrestricted time window
‒ Arbitrary number of independent CS measurements
‒ Arbitrary RF signals• Optical pulses and RF signal unsynchronized
• Pulses, chirps, sinusoids, communication waveforms …
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Photonic CS system using multimode waveguide
speckle
• Multimode waveguide/fiber replaces 2D SLM or WDM-Attenuator-WDMs
subsystem
• Exploit spatial randomness and wavelength sensitivity at output of
multimode guide
• Time-wavelength mapping multiplies RF signal by speckle wavelength
dependence
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Speckle Patterns at the output of a 1m, 105mm,
0.22NA step-index fiber
Speckle at each red dot uncorrelated with that at other dots
l = 1539.44 nm 1539.52nm
Small changes in wavelength can produce significant changes in speckle pattern
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Typical Rows in Speckle Measurement Matrix
Measured intensity as a function of wavelength for 4 locations in the output plane
of a 1m, 105mm, 0.22NA fiber
These patterns multiply the RF signal through time-wavelength mapping
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Simulated CS Results for Measured and Calculated
Measurement Matrices (MM)
Good agreement measured and calcuated MMs for multimode fiberSpeckle MMs as good as Gaussian RN MM
Four Measurement Matrices:
• Measured for multimode fiber (1m, 105mm, 0.22NA)
• Calculated from Gaussian random numbers with same mean and standard deviation as measured
• Calculated MM for multimode fiber with same dimensions as measured
• Calculated MM for Planar waveguide (10cm, 25mm, SOI)
Signal sparse under Identity Transform sparse under Haar Wavelet Transform
K=2 4 8 16K=2 4 8 16
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CS Probability of Recovery – 4 measurement matrices Signals sparse under identity transform
Sharp “phase transition” from no recovery to 100% recovery typical of good CS measurement matricesPlanar waveguide performance comparable to multimode fiber
Measured MM Gaussian RN MM
Multimode fiber
100% recovery0% recovery
Spars
ity
K
No. of Measurements M
Calculated MM Calculated MM
Multimode Fiber Planar Waveguide
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CS Probability of Recovery – measured fiber MM
Identity Discrete Cosine Haar Wavelet
Sp
ars
ity
K
Number of Measurements M
Slightly better performance for signals comprised of pulses (Identity transform)
Signals sparse under different transforms
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Demonstration of single channel of Speckle system
Non-uniform sampling given by speckle pattern dependence on wavelength/time
• 1-m multimode fiber‒ 105-mm core diameter, 0.22 NA, step-index
• Single photodiode placed in image plane of fiber output‒ PD diameter at fiber output = 14 mm, accounting for 72x image magnification
Periodic non-uniform Sampling with integration
ADC
Low-pass
Integrator
PD
Imaging Lens
1-m Multimode Fiber
Non-uniform sampling Pattern
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Experimental results for RF Chirped pulses
y
Time (Measurement No.)
• RF Parameters:
‒ Chirp = 20 MHz / 30 ms
‒ Carrier Freq = 2.453 GHz
‒ Maximum likelihood recovery
WDM and Speckle non-uniform sampling results nearly identical
Carrier Recovery vs. Number of Measurements
2446 2448 2450 2452 2454 2456 2458 24600
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25
30
35
40
Frequency, MHz
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Frequency, MHz
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Frequency, MHz
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Frequency, MHz
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ela
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80 272
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Next steps
• Multi-channel CS demonstration with multimode fiber
• Fabrication of Si planar waveguide
• Demonstration of CS with planar waveguide
• Integration of optical source, optical modulator, speckle waveguide
and photodiodes
• Demonstration of CS with integrated system
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Compressive Sensing Receiver Reticle Design
10cm multimode waveguide
2.0 cm
10cm multimode waveguide
terminated with single mode
input and 100 channel single
mode fanout
5cm multimode waveguide
terminated with single mode
input and 100 channel single
mode fanout
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Other applicatons of laser speckle
• Machine learning
– Saade et al. “Random projections through multiple optical scattering: Approximating kernels at
the speed of light” arXiv 2015
• Spectroscopy
– Redding and Cao. "Using a multimode fiber as a high-resolution, low-loss spectrometer." Optics
letters 2012 also Optics Express 2013
• Wavelength meter
– Mazilu et al. "Random super-prism wavelength meter." Optics letters 2014
• Strain sensor
– Varyshchuk,et al. "Using a multimode polymer optical fiber as a high sensitivy strain sensor."
2014.
• Imaging
– Liutkus et al. "Imaging with nature: Compressive imaging using a multiply scattering medium."
Scientific reports 2014.
– Kolenderska et al."Scanning-free imaging through a single fiber by random spatio-spectral
encoding." Optics letters 2015
– Shin et al. "Single-pixel imaging using compressed sensing and wavelength-dependent
scattering." Optics letters 2016.
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Conclusions
• Many undersampling strategies developed for measuring RF signals in the GHz band at
sub-Nyquist rates
• Intense work in the past 8 years on photonic implementations
– Most multiply an RF signal by a PRBS using an EO modulator
• Serial compressive sensing demonstrated with SLM applying PRBS
• Periodic non-uniform sampling demonstrated using
– WDM system
– Speckle in multimode waveguide
• Compressive sensing simulations
– Typical CS behavior with measured and calculated multimode fiber and calculated
multimode waveguide measurement matrices
– Sparsity/Measurement plots with sharp phase changes
• Simulations for planar waveguides indicate a path to a photonic integrated circuit for a CS
receiver in the GHz band