The document discusses measuring and utilizing transmission matrices to control light propagation through complex scattering media. It describes how the transmission matrix can be measured and used to focus light through scattering samples or transfer image information. Applications include focusing light to target areas and reconstructing images despite multiple scattering within biological tissues or other disordered materials.
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>>> Use
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Spatio-temporal control of light in complex media
1. Spatio-temporal control of
light in complex media
Sébastien
POPOFF
Directors : M. Fink et C. Boccara
Supervisors : S. Gigan et G. Lerosey
1
14/12/2011
2. Introduction
Imaging in optics
What are optical systems useful for?
Look further
Look smaller
14/12/2011 2
3. Introduction
Aberrations
Imaging in optics
Atmospheric
aberrations
14/12/2011 3
4. Introduction
Adaptive optics
Real-time correction of aberrations with adaptive optics
Courtesy: F. Lacombe/observatoire de Paris
Wavefront correction
(ex: deformable mirror)
Imaging
device
(CCD)
Wavefront Sensor
(ex: Hartmann-Schack)
Real-time control loop
14/12/2011 4
5. Introduction
Strong perturbations
AO convenient for wavefront perturbation :
Large spatial scale / small amplitude
Relevant for astronomy, free space optics, some biological applications…
What about stronger pertubations?
Multiple scattering, multiple reflections…
Techniques in Acoustics / Electromagnetism
Time reversal
Can we apply them in optics?
14/12/2011 5
6. Introduction
Time reversal
Time reversal mirror (Ultrasound experiment)
Hypothesis : linearity, reversibility of wave equation
Spatial and temporal focusing A. Derode, P. Roux et M. Fink, Phys. Rev. Lett., 75, 4206 (1995)
One-channel time reversal
importance of reflections
Temporal focusing Spatial focusing
14/12/2011 6
C. Draeger and M. Fink, Phys. Rev. Lett., 79, 407 (1997)
7. Introduction
Time reversal
If no access to temporal details
Monochromatic counterpart of TR: Phase conjugation
Reverse time conjugate the phase
Spatial focusing
14/12/2011 7
8. Introduction
New techniques of light control
What about optics?
Spatial light modulators (SLM) Temporal control:
- Pulse shaping
- Modulators
Acousto-optic modulators (up to GHz)
Electro-optic modulators ( > 10 GHz)
Allow a high degree of control
on light propagation!
Deformable mirrors: up to 4000 elements –
kHz – expensive
Liquid cristals technology: ~1 million pixels –
~100Hz – cheap
14/12/2011 8
9. Outline
I. Transmission matrix in scattering media
II. Reflection matrix and optical “DORT”
III. Complex envelope time reversal
14/12/2011 9
10. Transmission matrix in scattering media
Introduction
In every day life…
…clouds… …white paint…
…biological tissues !
14/12/2011 10
11. Transmission matrix in scattering media
Scattering: complex but coherent process
Simple case
Young slits:
Fringes : Two waves interference
Thick disordered media:
Speckle
- Multiple events of diffusion
- Position of diffuser unknown
14/12/2011 11
12. Transmission matrix in scattering media
Multiple scattering: too complex
White paint
100μm (particle size ≤ 1 μm)
>108 particles
Impossible to simulate
1mm²
Only predictions accessible: Mesoscopic physics
Statistical properties on transport, correlations, fluctuations
No knowledge of the field for a given realization
14/12/2011 12
13. Transmission matrix in scattering media
A pioneering experiment
A speckle grain:
• Interference of a great number of optical paths
Sum of terms of random phases (phasors)
• Contributions in phase constructive
interferences of multiple paths
14/12/2011 13
15. Transmission matrix in scattering media
Improve the resolution
λf1/D1
λf2/D2
Acoustics: A. Derode, P. Roux and M. Fink , Phys. Rev. Lett., 75, 4206 (1995)
Optics: I. M. Vellekoop, A. Lagendijk and A. P. Mosk, Nature Photonics 4, 320 - 322 (2010)
14/12/2011 15
16. Transmission matrix in scattering media
First experiment
Remarks:
- 1 optimization = 1 focal spot
Need to optimize for each target
- Optimization: only indirect information on the medium
Can we go further?
14/12/2011 16
17. Transmission matrix in scattering media
Basic principle
SLM : array of pixels Linear system CCD camera : array of pixels
= = =
14/12/2011 17
18. Transmission matrix in scattering media
Linear media and matrices
E in Input field 1.. N
E out
m
in
hmn En E out H .E in
E out Output field n
Input k
Output k
Free space
Direct access to
information
Identity Matrix
Input k
Output k
Scattering sample
Information shuffled but
not lost!
Seemingly Random Matrix
14/12/2011 18
19. Transmission matrix in scattering media
Setup
Objective : Measuring the Transmission Matrix
Hypothesis : Coherence of the illumination, Stability of the Medium, Linearity
Sample
ZnO
L = 80 25
Output μm l* = 6 2 μm
Detection
(Interferometry)
1 macropixel ↔
k vector
Input Control
Spatial Light Modulator
(SLM) in Phase Only
Modulation
macropixel ↔ k vector
14/12/2011 19
20. Transmission matrix in scattering media
Measurement of the Transmission Matrix
Step by step reconstruction
1..N
out in
E m hmn En
n
Pixel off Pixel on
In practice, we use φ=+π/2
Hadamard vectors φ=-π/2
(Phase-only SLM,SNR) , , , etc…
E. Herbert, M. Pernot, G. Montaldo, M. Fink and M. Tanter, IEEE UFFC, 56, 2388, (2009)
14/12/2011 20
21. Transmission matrix in scattering media
Construction of the Transmission Matrix
Transmission matrix
(filtered to remove effect of the reference)
14/12/2011 21
22. Transmission matrix in scattering media
Applications: Focusing
What can we do with the TM?
Calculate the mask to display!
CCD
SLM
sample
Only one measurement of
the TM
CCD
SLM
sample
CCD
SLM
Plane wave illumination
14/12/2011 sample 22
23. Transmission matrix in scattering media
Applications: Focusing
Which mask to focus?
Phase conjugated mask
Put contributions in phase on one
?
spot ↔ A. Mosk experiment
E in t *
H E target E out H t H *.E target
Strong values in the diagonal
We can focus everywhere
N=256
t *
H H
Non-diagonal elements not zero
Imperfection inherent to PC
14/12/2011 N=256 modes (16x16 pixels on the CCD) 23
24. Transmission matrix in scattering media
Can we go beyond phase conjugation?
Statistical properties of the TM
Transfer of information (image)
14/12/2011 24
25. Transmission matrix in scattering media
Statistical properties of the transmission matrix
Tool: Singular Value Decomposition
(generalization of diagonalization for Output basis
any Matrix) H U V* Input basis
0 0 0 - i >0 represents the amplitude transmission
1
through the ith channel.
0 2 0 0
0 0 ... ...
-Σλi2 corresponds to the total transmittance for a
0 0 ... N plane wave
We study the distribution of (normalized) singular values ρ(λ)
14/12/2011 25
26. Transmission matrix in scattering media
Statistical properties of the transmission matrix
A general Random Matrix Theory prediction : quarter circle law distribution
Transmission matrix
(filtered to remove effect of the
reference) In acoustics:
A. Aubry et al., Phys. Rev. Lett., 102, 84301, (2009)
Signature of randomness
14/12/2011 26
27. Transmission matrix in scattering media
Applications: Image transmission
CCD
SLM
sample
? TM
We want OH
E img O.E out OH .E obj close to Identity
Finding Eobj knowing Eout Shaping
14/12/2011 27
28. Transmission matrix in scattering media
Applications: Image transmission
What operator to reconstruct a complex image? (knowing the TM)
1 Perfect reconstruction
Inversion : O H OH I
Not stable in presence of noise
1 0 0 0 1/ 1 0 0 0
0 0 0 0 1/ 2 0 0
2
low λi high 1/λi
0 0 ... ... 0 0 ... ...
If noise, H-1 dominated by noise !
0 0 ... N 0 0 ... 1/ N
14/12/2011 28
29. Transmission matrix in scattering media
Applications: Image transmission
What operator to reconstruct a complex image ?
t * Very stable
Phase Conjugation : O H Reconstruction perturbated when the
t
OH H *H image is complex
t
H λi λi H*
t
H *H
N=100
14/12/2011 N=100 29
30. Transmission matrix in scattering media
Applications: Image transmission
t * 1 t
A tradeoff : Tikhonov Regularization O H .H I H*
(A.N.Tikhonov, Soviet. Math. Dokl., 1963)
0 (Noiseless) (Noisy)
1 t *
O H O H
Optimal Operator for σ = Noise variance
14/12/2011 30
31. Transmission matrix in scattering media
Applications: Image transmission
Experimental Results :
Output Speckle (Eout)
Input Mask (Eobj)
Inversion Phase Conjugation Regularization
Reconstruction
C = 11% C = 76% C = 95%
14/12/2011 31
33. Transmission matrix in scattering media
Conclusion and Perspective
We did:
- Focusing and information transfer through complex medium
- Studied statistical properties of a scattering medium
More:
- Develop a faster setup (micromirror arrays, ferromagnetic SLMs) for
biological purposes
- Study more complex media (Anderson localization, photonic
cristals…)
References :
- S.M. Popoff, G. Lerosey, R. Carminati, M. Fink, A.C. Boccara and S. Gigan, Phys. Rev. Lett 104, 100601, (2010)
- S.M. Popoff, G. Lerosey, M. Fink, A.C. Boccara and S. Gigan, Nat. Commun., 1,1 ncomms1078 (2010)
Related papers :
- I.M. Vellekoop and A.P. Mosk, Opt. Lett. 32, 2309 (2007).
-Z. Yaqoob, D. Psaltis, M.S. and Feld and C. Yang, Nat. Phot., 2, 110 (2008).
And many many more !
14/12/2011 33
35. From transmission matrix to reflection matrix
SLM : array of pixels
Linear sample
CCD camera : array of pixels
14/12/2011 35
36. Reflection matrix and optical “DORT”
I. Transmission matrix in scattering media
II. Reflection matrix and optical “DORT”
III. Complex envelope time reversal
14/12/2011 36
37. Reflection matrix and optical “DORT”
Introduction
Applications of the RM for multiply scattering media?
Measure of the CBS cone as in acoustics
Optics: M.P.V. Albada and A. Lagendijk, Phys. Rev. Lett., 55,2692 (1985)
Acoustics: A; Tourin et al, Phys. Rev. Lett., 79, 3637, (1997)
A. Aubry et al., Phys. Rev. Lett., 102, 84301, (2009)
Problem:
Measurement in optics: noise, specular reflections…
Application in freespace / aberrating medium (simple scattering):
The DORT method in optics (suggested by A. Aubry)
14/12/2011 37
38. Reflection matrix and optical “DORT”
Introduction
E0 KE0
Iterative time reversal
K * E0
*
KK * E0
*
K * KE0 KK * KE0
14/12/2011 38
39. Reflection matrix and optical “DORT”
Introduction
At step n:
2n t * n
E K K E0 0
1 0 0
0
0
0
2
0 0 ... ...
SVD of K: 0 0 ... N
Output basis
K U V* Input basis 1
2n
0 0 0
2n 0 0 0 0
0 0 ... ...
2n 2n * 0 0 ... 0
E U V E0 2n 2n 2n
1 2 ... N
14/12/2011 39
40. Reflection matrix and optical “DORT”
Introduction
1 strong singular value ↔ 1 scatterer ?
DORT:
- Mesure of the RM
- SVD of the RM
- Display the first singular vectors
14/12/2011 40
41. Reflection matrix and optical “DORT”
Introduction
Works with an aberrating medium
(single scattering only)
Hypothesis: linearity, single scattering regime
14/12/2011 41
42. Reflection matrix and optical “DORT”
Setup
Scatterers:
100 nm isotropic gold
particles on a glass
slide
Cross Polarization
Control
Aberrating
medium
14/12/2011 42
43. Reflection matrix and optical “DORT”
Problems
The energy measured should only come from the scatterers
Problem:
- Important contributions of specular reflections !
Solutions:
Pin
- Cross polarization
x
- (Dark field) k
y
100 nm gold
x beads
k
Pout
y
14/12/2011 43
44. Reflection matrix and optical “DORT”
Selective Focusing
Reflection
Control
14/12/2011 44
45. Reflection matrix and optical “DORT”
Setup
Scatterers:
100 nm isotropic gold
particles on a glass
slide
Cross Polarization
Aberrating
medium
14/12/2011 45
46. Reflection matrix and optical “DORT”
Adaptive optics
Aspect of the first input singular vector (phase mask)
Free space ~ lens With aberrating mediums
14/12/2011 46
47. Reflection matrix and optical “DORT”
Modes of a single particles
Particle ~
3 orthogonal dipoles
Need for sufficient NA to
excite the dipoles with one
input polarization
y component of the
output field
14/12/2011 47
48. Reflection matrix and optical “DORT”
Modes of a single particle
Theoretical singular value distribution
(vector diffraction theory)
Number of SV
Py Dipole Pz Dipole
Px Dipole
14/12/2011 48
50. Reflection matrix and optical “DORT”
Modes of a single particle
Experimental singular value distribution
? Pz dipole Px dipole
Number of SV
Py Dipole Pz Dipole
Px Dipole
14/12/2011 50
51. Reflection matrix and optical “DORT”
Conclusions and Perspectives
We did:
- Selective focusing through aberrating medium
- Radiation pattern analysis of a single nanobead
More:
- Reduce specular reflections (dark field)
- Develop a setup more stable (laser)
Pattern analysis for characterization, plasmonic, …
References :
- S.M. Popoff, A. Aubry ,G. Lerosey, M. Fink, A.C. Boccara and S. Gigan, Phys. Rev. Lett. (in press)
14/12/2011 51
52. Complex envelope time reversal
I. Transmission matrix in scattering media
II. Reflection matrix and optical “DORT”
III. Complex envelope time reversal
14/12/2011 52
53. Complex envelope time reversal
Spatio-temporal focusing in complex media
With spatial degrees of freedom With temporal degrees of freedom
(pulse shaping)
J. Aulbach et al., Phys. Rev. Lett., 106,103901 (2011)
O. Katz et al., Nat. Photonics, 5, 372, (2011) D. McCabe et al., Nat Commun., 2, 447, (2011)
14/12/2011 53
54. Complex envelope time reversal
Modulation for telecommunications
When only low frequencies accessible
Modulation (Telecomunications)
Carrier wave Signal
x
= Propagation
Detector
Independent modulation in phase and quadrature (IQ)
Use high frequency waves with ‘low’ frequency generator / oscilloscope
Lower bandwidth but very high spectral resolution
Modulators and demodulators widely available for telecommunications ($$$)
14/12/2011 54
55. Complex envelope time reversal
Time reversal
hAB (t ) E (t ).e j t
hAB ( t ) E ( t ).e j t -t
Spatial and temporal focusing TR = reverse modulation
+ conjugate carrier wave
G. Lerosey et al., Phys. Rel. Lett., 92, 193904 (2004)
Time (μs) Time (μs)
Pulse in modulation at A (on one quadrature) Signal received at A after time reversal
14/12/2011 55
58. Complex envelope time reversal
Bandwidth vs medium’s correlation frequency
Lifetime in system need to be >> 1/Δf modulation
Electromagnetism experiment:
Huge cavity needed ( > 13m3) Huge number of modes (λ2.45GHz = 12cm )
Impulse response B
B
A
Time (μs)
G. Lerosey et al., Phys. Rev. Lett., 92, 193904 (2004)
Same problem in optics
Need for strong dispersion / strong enough signal
14/12/2011 58
59. Complex envelope time reversal
Temporal focusing
Looped single
mode cavity
input output
Evanescent
coupling
Channel I Channel Q
Impulse
response
Numerical
time reversal
(correlations)
14/12/2011 59
60. Complex envelope time reversal
Temporal focusing
Channel I Channel Q
Numerical
time reversal
(correlations)
Experimental
time reversal
Demonstration of the compression of the impulse
response by time reversal
Application : fiber optics telecommunication
14/12/2011 60
61. Complex envelope time reversal
Towards spatio-temporal focusing
Problems : Weak signals / Need for very strong dispersion
Multimode fiber cavity
Scattering medium
input output
Chaotic
3D cavity
Still in progress!
14/12/2011 61
62. Conclusion
I. Transmission matrix in scattering media
- Spatial focusing
- Image transmission
- Singular value analysis
II. Reflection matrix and optical “DORT”
- Selective focusing through an aberrating medium
- Scattering pattern analysis
III. Complex envelope time reversal
- Temporal focusing
- Towards spatial and temporal focusing...
14/12/2011 62
63. Remerciements :
Collaborateurs : Préparation échantillons :
Laurent Boitard
Sylvain Gigan
Gilles Tessier
Geoffroy Lerosey
Benoit Malher
Alexandre Aubry
Olivier Loison
Remi Carminati
Mathias Fink Aide au montage :
Claude Boccara Aurélien Peilloux
Théorie : Sébastien Bidault
Samuel Grésillon Caractérisation des échantillons :
Support divers : Matthieu Leclerc
Marie Lattelais
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65. Transmission matrix in scattering media
Statistical properties of the transmission matrix
Hobs H. ref
Artefact :
« raster » effect
due to the
amplitude of Sref
Effect of ref
Observed Matrix
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66. Transmission matrix in scattering media
Setup
Objective : Measuring the Transmission Matrix
Hypothesis : Coherence of the illumination, Stability of the Medium, Linearity
Input Control
Spatial Light Modulator
(SLM) in Phase Only
Modulation
Output A macropixel ↔ A k
Detection vector
CCD Camera
A macropixel
↔ A k vector
Sample
ZnO
L=
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67. Matrice de Transmission Optique d’un Milieu Diffusant
Applications : Transmission d’Image
Efficacité de la reconstruction en fonction
de σ
σ
Filtrage inverse Filtrage adapté
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68. Matrice de Transmission et Milieu Diffusant
Propriétés Statistiques de la Matrice de Transmission
Une prédiction générale des matrices aléatoires : “Loi du quart de cercle”
obs
fil hmn
hmn obs
hmn
m
Matrice Observée Matrice Filtrée
Filtrage de Hobs pour éliminer les effets de la référence
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69. Transmission matrix in scattering media
Applications : Focusing
Theoretical focusing
VS
Experimental focusing
Target Expected focusing from Experimental
measured matrix focusing
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70. Transmission matrix in scattering media
Stability and Measurement Time
TM Measurement Time ~ 15 min
(1024x1024 )
Decorrelation Time of ~ 1 hour
ZnO deposit
Decorrelation Time of
<< 1s
Biological Tissues
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71. Reflection matrix and optical “DORT”
Introduction
The reflection matrix
En n
E in Input field
E out Output field
1..N
out in
Em kmn En
k mn En m n
E out K .E in
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72. Optical time reversal in modulation
Time reversal
Time reversal in modulation Signal received at B
B
A TR = reverse modulation
+ conjugate carrier wave
Pulse in modulation at A (on one quadrature)
Signal received at A
Spatial and temporal focusing
G. Lerosey et al., Phys. Rel. Lett., 92, 193904 (2004)
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73. Transmission matrix in scattering media
The matrix model : A conveniant model
Free space Multiply scattering sample
Detrimental to Conventional Optical Techniques
Matrix Description to link input / output k vectors
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74. Transmission matrix in scattering media
Measuring the Complex Output Field
2
I out Eout No phase information !
i 2
I Eout e Eref Interferometric stability
for several minutes !
E ref uniform
3 1
0 i
Eout I I i I 2
e I 2
3 1
E ref not uniform
I0 I i I 2
ei I 2
*
OK as long as …..
Eout .Eref …. is constant
E ref
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75. Transmission Matrix of an Optical Scattering Medium
Theoretical Focus Spot
λf1/D1
λf2/D2
I. M. Vellekoop, A. Lagendijk & A. P. Mosk, Nature Photonics 4, 320 - 322 (2010)
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76. Transmission Matrix of an Optical Scattering Medium
Theoretical Focus Spot
D λF/D SLM L λl/L
l
F
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77. Complex envelope time reversal
Time reversal
Time reversal in modulation in Signal received at B
a reverberant cavity
B
TR = reverse modulation
A + conjugate carrier wave
Pulse in modulation at A (on one quadrature)
Signal received at A
Spatial and temporal focusing
G. Lerosey et al., Phys. Rel. Lett., 92, 193904 (2004)
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78. Transmission matrix in scattering media
Linear media and matrices
E in Input field 1.. N
E out
m
in
hmn En E out H .E in
E out Output field n
Input k
Free space
Output
Direct access to
k
information
Identity Matrix
Input k
Scattering sample
Output
Information shuffled but
k
not lost !
Seemingly Random Matrix
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