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.

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
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3. Introduction
Aberrations
Imaging in optics
Atmospheric
aberrations
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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
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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?
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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
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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
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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
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9. Outline
I. Transmission matrix in scattering media
II. Reflection matrix and optical “DORT”
III. Complex envelope time reversal
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10. Transmission matrix in scattering media
Introduction
In every day life…
…clouds… …white paint…
…biological tissues !
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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
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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
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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
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14. Transmission matrix in scattering media
A pioneering experiment
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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)
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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?
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17. Transmission matrix in scattering media
Basic principle
SLM : array of pixels Linear system CCD camera : array of pixels
= = =
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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
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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
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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)
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21. Transmission matrix in scattering media
Construction of the Transmission Matrix
Transmission matrix
(filtered to remove effect of the reference)
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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
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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
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24. Transmission matrix in scattering media
Can we go beyond phase conjugation?
 Statistical properties of the TM
 Transfer of information (image)
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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)
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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
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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
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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
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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
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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
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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%
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32. Transmission matrix in scattering media
Applications: Image transmission
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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 !
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34. From transmission matrix to reflection matrix
SLM
Linear sample CCD camera
= = =
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35. From transmission matrix to reflection matrix
SLM : array of pixels
Linear sample
CCD camera : array of pixels
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36. Reflection matrix and optical “DORT”
I. Transmission matrix in scattering media
II. Reflection matrix and optical “DORT”
III. Complex envelope time reversal
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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)
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38. Reflection matrix and optical “DORT”
Introduction
E0 KE0
Iterative time reversal
K * E0
*
KK * E0
*
K * KE0 KK * KE0
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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
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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
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41. Reflection matrix and optical “DORT”
Introduction
Works with an aberrating medium
(single scattering only)
Hypothesis: linearity, single scattering regime
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42. Reflection matrix and optical “DORT”
Setup
Scatterers:
100 nm isotropic gold
particles on a glass
slide
Cross Polarization
Control
Aberrating
medium
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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
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44. Reflection matrix and optical “DORT”
Selective Focusing
Reflection
Control
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45. Reflection matrix and optical “DORT”
Setup
Scatterers:
100 nm isotropic gold
particles on a glass
slide
Cross Polarization
Aberrating
medium
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46. Reflection matrix and optical “DORT”
Adaptive optics
Aspect of the first input singular vector (phase mask)
Free space ~ lens With aberrating mediums
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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
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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
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49. Reflection matrix and optical “DORT”
Modes of a single particle
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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
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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)
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52. Complex envelope time reversal
I. Transmission matrix in scattering media
II. Reflection matrix and optical “DORT”
III. Complex envelope time reversal
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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)
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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 ($$$)
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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
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56. Complex envelope time reversal
Setup
Setup
Modulation Part: Demodulation Part:
- 10 GHz arbitrary waveform generator - Interferometric detection of 2 quadratures
- Triple Mach-Zehnder modulator - 50 GHz oscilloscope
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57. Complex envelope time reversal
Modulation / Demodulation
Modulation Part:
- 10 GHz arbitrary waveform generator
- Triple Mach-Zehnder modulator (Photline)
Demodulation Part:
- Interferometric detection of 2 quadratures
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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
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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)
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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
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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!
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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...
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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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64. Collaborateurs
Sylvain Geoffroy Alexandre Rémi Mathias Claude
GIGAN LEROSEY AUBRY CARMINATI FINK BOCCARA
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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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