Transcript PPT

Sparsity-based sub-wavelength imaging
and super-resolution in time and frequency
Yoav Shechtman
Physics Department, Technion, Haifa 32000, Israel
Alex Szameit, Snir Gazit, Pavel Sidorenko, Elad Bullkich, Eli Osherovic,
Michael Zibulevsky, Irad Yavneh, Yonina Eldar , Oren Cohen, Moti Segev
FRISNO 2011
Nonlinear Optics Laboratory
Sub-wavelength images in the microscope
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Optical cut-off for high spatial frequencies
field propagation (z = 0 → z > 0)
2
iz  2   k x2  k y2  

 ( x, y, z )  FT  FT  ( x, y, z  0)e



1
k k 
2
x
2
y
k k 
2
x
2
y
2

2

propagating waves
evanescent waves
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Hardware solutions for sub-wavelength imaging
• Scanning near-field optical microscope
• Methods using florescent particles
• Structured Illumination
• Negative-index / metamaterials structures: superlens, hyperlens
• Hot-spot methods: nano-hole array, super-oscillations
Require scanning, averaging over multiple experiments
Is it possible to have real-time, single exposure subwavelength imaging using a ‘regular’ microscope?
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Analytic Continuation
• The 2D Fourier transform of a spatially bounded function is an
analytic function.
(k x , k y )  FT ( x, y, z  0)eikz z
• Problem: Existing analytic continuation methods are not very
robust:
• sampling theorem based extrapolations yield a highly ill posed
matrix.
• Iterative methods (Gerchberg - Papoulis) are sensitive to noise
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Common wisdom
“All methods for extrapolating bandwidth beyond the diffraction
limit are known to be extremely sensitive to both
• noise in the measured data and
• the accuracy of the assumed a priori knowledge.”
“It is generally agreed that the Rayleigh diffraction limit represents
a practical frontier that cannot be overcome with a conventional
imaging system.”
J. W. Goodman, Introduction to Fourier Optics, 2005
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Bandwidth extrapolation problem:
infinite number of possible solutions!
Measurements
How to choose the right one?
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Problem: non-invertible filter
FT(signal)
Measurements
   
  
 y4  
y  
 3 
 y2  
 y  
 1 
 y0    
  
 y1  
 y 2  
  
 y 3  
y  
  4  
   
signal
FilteredFourier
Fouriertransform
transform(invertible)
(non-invertible)

A 4 4
A3 4
A 2 4
A1 4
A0 4
A1 4
A2 4
A3 4
A4 4
A 43
A33
A 23
A13
A03
A13
A23
A33
A43
A 4 2
A3 2
A 2 2
A1 2
A0 2
A1 2
A2 2
A3 2
A4 2
A 41
A31
A 21
A11
A01
A11
A21
A31
A41
A 40
A30
A 20
A10
A00
A10
A20
A30
A40

A 41
A31
A 21
A11
A01
A11
A21
A31
A41
A 42
A32
A 22
A12
A02
A12
A22
A32
A42
A 43
A33
A 23
A13
A03
A13
A23
A33
A43
A 44
A34
A 24
A14
A04
A14
A24
A34
A44
  
 
 x4 
 x 
 3 
 x2 
 x 
 1 
 x0 
 
 x1 
 x 2 
 
 x3 
 x 
  4 
  
buried in the noise
(exponentially small evanescent waves)
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• Under-determined system of equations:
more variables than equations
Ax  y
• Infinite number of solutions (x)
• Choose the one that “makes the most sense”
We• choose
thewhat?
solution with maximum sparsity –
Based on
theBased
one with
fewest
on the the
knowledge
thatnonzero elements.
the object is sparse in a known basis.
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Why sparsity?
• General: Many objects are sparse in some
(general) basis.
• Powerful:
• Robust to noise.
Without noise, in a sparse enough case
the sparsest solution is unique
• Sparsity is used successfully for image
denoising, deconvolution, compression,
enhancement of MR images and more.
However – has never been used for sub-λ
imaging, or temporal bandwidth extrapolation.
• Attainable: Efficient algorithms exist for estimating
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the sparsest solution.
10
Sparsity – a general feature of information
Sparsity in real space image
biological species:
Real-space sparsity ~ 2- 5%
Sparsity in another basis
Electronic chips:
Sparsity in gradient basis ~ few %
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How to do it: for example - Basis Pursuit
Solve the (convex) optimization problem:
•
x :unknown image
•
y: measured image
•
A:Low-pass filter + sparsity basis
•
ε: Noise parameter
•
The requirement on the l1 norm is to promote sparsity.
•
Find the sparsest x that is consistent with the measurements.
[S.S. Chen et al., SIAM Journal on Scientific Computing, 20, 33 (1998)]
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Proof of concept
diffuser
(optional)
original
filtered image
reconstructed
image
Gazit et al., Opt. Exp. Dec. 2009
Shechtman et al., Opt. Lett. Feb. 2010
tunable
filter
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Proof of concept
Original
Recovered
Gazit et al., Opt. Exp. Dec. 2009
Shechtman et al., Opt. Lett. Feb. 2010
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True sub- experiments – 1D @  = 532 nm
SEM image
Chromium
Glass
Width: 150 nm
Length: 20 µm
Spacing:
150 nm (left/right pair)
300 nm (center) ~ diffraction
limit
Fabrication:
Kley – group
University of Jena
[email protected]
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Best possible microscope image (NA ≈ 1)
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Microscope image far-field
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Experimental result (with hand-made microscope)
reconstruction
microscope
image
150 nm
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Comparison original - reconstruction
real space
spatial spectrum
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True sub- experiments – 2D @  = 532 nm
100 nm
100 nm
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Best possible microscope image (NA ≈ 1)
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Microscope image far-field
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Loss of power in the far-field
more than 90% of the intensity is lost
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Experimental results
microscope image
reconstructed
image
SEM image
100 nm
Abbe limit
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Sub- imaging from far-field intensity measurements
Can we do sub-wavelength reconstruction
based on intensity measurements only?
Without measuring phase at all?
Yes, indeed. The knowledge of sparsity is powerful.
First:
Fourier phase recovery using iterative algorithm* –
given the blurred image intensity and Fourier intensity.
Second: sparsity-based reconstruction using recovered phase.
or, better, combine the two!
* J. R. Fienup, Appl. Opt. 21 (1982)
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Experimental: sparsity-based recovery
of ‘random’ distribution of circles
Sparse recovery *
Blurred image
SEM image
Circles are
100 nm
diameter
462
464
50
466
100
468
470
150
472
200
474
476
250
474
476
478
480
482
484
50
486
100
150
200
250
Diffraction-limited
* Assuming non-negativity
(low frequency)
Model
intensity measurements Fourier transform
Measured (phase part)
Model FT intensity
-6
10
20
-4
30
Frequency [1/ ]
Wavelength
~ 532 nm
40
50
60
70
-2
0
2
80
4
90
10
20
30
40
50
60
70
80
90
6
-5
0
Frequency [1/ ]
5
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Experimental: incorrect reconstruction
with wrong number of circles
30 circles left
22 circles left
11 circles left
12 circles left
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Sparsity-based super-resolution in pulse-shape measurements – experimental
Slow Photodiode (τ~ 1 ns)
Laser
Pulse
Fast Photodiode
(τ~ 175 ps)
Vosc (t )   IRF (t  t ' ) I Laser (t ' )dt ' Vosc ( )  T ( ) I Laser ( )
1
(b)
100
Slow PD
Fast PD
0.8
0.6
0.4
0.2
0
-1.5
-1
-0.5
0
0.5
Time [ns]
1
Transfer functions
Intensity [a.u.]
Intensity [a.u.]
Impulse response functions
1.5
(c)
10-2
10-4
Slow PD
Fast PD
0.1
1
10
Frequency [GHz]
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Sparsity-based super-resolution in pulse-shape measurements – experimental
Spectra of measured signals
Fast PD
Slow PD
100
Intensity [a.u.]
VOSC [a.u.]
1
Measured signals
0.8
0.6
0.4
0.2
(c)
10-2
10-4
Fast PD
Slow PD
0
-1.5
-1
-0.5
0
0.5
1
1.5
0.1
1
10
Frequency [GHz]
Time [ns]
1
0.8
100
Deconv Fast PD
Reconstructed
Deconv Slow PD
Intensity [a.u.]
Intensity [a.u.]
Reconstruction
0.6
0.4
0.2
0
-1.5
-1
-0.5
0
0.5
Time [ns]
1
1.5
(c)
10-2
10-4
Deconv Fast PD
Reconstructed
Deconv Slow PD
0.1
1
10
Frequency [GHz]
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Sparsity-based super-resolution FTIR
spectroscopy
-x
0
x
Light Source
Power (W)
Fixed Mirror
BS
Moving Mirror
Power (W)
X (cm)
λ (nm)
Detector
Because the interferogram cannot be collected from x = - to +,
it is always truncated,
hence some error arises in the resulting spectrum:
the spectral line is broadened + side-lobes are added
Resolution of a F-T spectrometer:
Δλ = 1 / (path difference = 4x)
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FFT
0.01
0
-0.01
-0.02
-0.03
-200
Intensity [a.u.[
0.03
-100
0
100
OPD [um]
200
Truncated Interferogram
0.02
FFT
0.01
0
-0.01
-0.02
-0.03
-200 -100
0
OPD [um]
100
200
Spectrum
1
0.8
0.6
0.4
0.2
0
700
750
800
Wavelength
[nm]
Spectrum
850
900
1
0.8
0.6
0.4
Spectral Intensity normalized
Intensity [a.u.[
0.02
Spectral Intensity normalized
Full Interferogram
0.03
Spectral Intensity normalized
Sparsity-based super-resolution in FTIR spectrum measurement – experimental example
Spectrum
CS
Full FTIR
1
0.8
Truncated
Interferogram
0.6
0.4
0.2
0
750
800
850
Wavelength [nm]
0.2
0
700
750
800
Wavelength
[nm]
850
900
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Conclusions
• method for recovering sub- information
from the optical far-field of images
• requires no additional hardware
• works in real time and with ultrashort pulses
• applicable to all microscopes (optical and non-optical)
• reconstruction also with incoherent / partially coherent light
• Ideas are universal: can be used to recover
•
shapes of ultrashort pulses in time
•
spectral features
quantum info!
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Many thanks
for your attention!
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A little about uniqueness
An object comprising on n ‘features’ is uniquely determined by 2n(n+1)
measurements on a polar grid in k-space, without noise.
Y. Nemirovsky, Y. Shechtman, A. Szameit, Y.C. Eldar, M. Segev, in preparation
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Comparison of approaches
Original CS approach
Our CS-related approach
• measurement in uncorrelated basis
(commonly Fourier basis)
• measurement in far-field
(= Fourier basis)
OR blurred near field
or in between
• sampling (randomly) over the
entire measurement basis with
low resolution
• sampling in a small part of the
measurement basis (kx < k) with
high resolution
• reduction of required samples
to retrieve the function
• obtain maximal info on the
frequency region where we
do NOT measure
We do NOT do CS. We do NOT use CS “rules”.
Why does it work for us?
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Unique sparse solution
sparse
y  Wd1  Wd 2
W (d1  d2 )  Wz  0
triangle inequality:
d1  d 2 0  d1 0  d 2
Wz  0
0
if every S1+S2 columns
of W are linearly independant
z 0  S1  S2
z  0 z
d1  d 2
1
1 
 , then there is a unique sparse solution
if d 0  1 
2   (W ) 
matrix coherence
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Reconstruction of the phase (Fienup-Algorithm)
real space
far-field
Iteration
phase
Fienup, Opt. Lett. 3, 27 (1978).
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Experimental: holes on a grid
139 nm
Model phase
3
20
2
40
1
60
0
80
-1
100
-2
-3
Recovered image
20
1
-1
40
60
80
100
Recovered phase
-1
3
0.8
-0.5
2
y [a.u.]
-0.5
1
0.6
0
0
0
0.4
-1
0.5
0.5
1
-1
-2
0.2
1
-0.5
0
x [a.u.]
0.5
1
0 -1
-3
-0.5
0
0.5
1
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* Consider a function f  t   N that can be written as a superposition of
spikes: f  t    f     t   
 T
If it is comprises of T spikes, and N is a prime number, then f  t  can be
uniquely defined by any 2 T of its Fourier measurements, defined as:
N 1
f     f  t  e  i 2t / N
  0,1.... N  1
t 0
Specifically, the 2 T low pass Fourier coefficients will do.
* Candes,
E.J.; Romberg, J.; Tao, T.; , "Robust uncertainty principles: exact signal reconstruction from
highly incomplete frequency information," Information Theory, IEEE Transactions on , vol.52, no.2, pp.
489- 509, Feb. 2006
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Sparsity-based super-resolution in pulse-shape measurements – theoretical example
Source laser pulse
Oscilloscope signal
1
VOSC
Intensity
1
0
0
0
50
100 150
0
Intensity
time [ps]
800
time [ps]
100
100
10-2
10-5
10-4
400
IRF
10-10
10-6
10-15
VOSC
10-8
10-10
10-20
1
10
100
Freq. [GHz]
1
10
100
Freq. [GHz]
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Sparsity-based super-resolution in pulse-shape measurements – theoretical example
Source laser pulse
Oscilloscope signal
1
VOSC
Intensity
1
0
0
0
50
100 150
0
Intensity
time [ps]
800
time [ps]
100
100
10-2
10-5
10-4
400
IRF
10-10
Without noise
De-convolution  perfect reconstruction
I Laser ( ) 
Vosc ( )
T ( )
10-6
10-15
VOSC
10-8
10-10
10-20
1
10
100
Freq. [GHz]
1
10
100
Freq. [GHz]
Nonlinear Optics Laboratory
Sparsity-based super-resolution in pulse-shape measurements – theoretical example
Source laser pulse
Oscilloscope signal
Wiener Deconvolution
1
1
Intensity
VOSC
Intensity
1
0
0
50
100 150
0
Intensity
time [ps]
800
100
10-2
10-5
100
10-15
VOSC
10-8
10-20
10
100
Freq. [GHz]
10-2
IRF
10-10
1
100 150
time [ps]
10-6
10-10
50
time [ps]
100
10-4
400
1
10
100
Freq. [GHz]
Intensity
0
0
10-4
10-6
10-8
10-10 1
10
100
Freq. [GHz]
Nonlinear Optics Laboratory
Sparsity-based super-resolution in pulse-shape measurements – theoretical example
Source laser pulse
Oscilloscope signal
Wiener Deconvolution
Sparsity-based reconstruction
1
1
0
50
100 150
0
Intensity
time [ps]
100
10-2
10-5
VOSC
10-8
10-20
100
Freq. [GHz]
50
1
10
100
Freq. [GHz]
100 150
time [ps]
100
10-2
IRF
10-15
10
100 150
100
10-10
1
50
time [ps]
10-6
10-10
Intensity
800
time [ps]
100
10-4
400
0
Intensity
0
0
Intensity
0
1
Intensity
VOSC
Intensity
1
10-4
10-6
10-5
10-10
10-8
10-10 1
10
100
Freq. [GHz]
1
10
100
Freq. [GHz]
• 40 ps features are well reconstructed (τ~1 ns)
• Resolution is enhanced by >10 times vs. Wiener de-convolution
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