Spatio-temporal control of light in complex media

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

Introduction
Imaging in optics

What are optical systems useful for?
Look further

Look smaller

14/12/2011 2

Introduction
Aberrations

Imaging in optics

Atmospheric
aberrations

14/12/2011 3

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

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

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)

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

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

Outline

I. Transmission matrix in scattering media
II. Reflection matrix and optical “DORT”
III. Complex envelope time reversal

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Transmission matrix in scattering media
Introduction

In every day life…

…clouds… …white paint…

…biological tissues !

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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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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

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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A pioneering experiment

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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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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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Basic principle

SLM : array of pixels Linear system CCD camera : array of pixels

= = =

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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

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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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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Construction of the Transmission Matrix

Transmission matrix
(filtered to remove effect of the reference)

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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

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


Can we go beyond phase conjugation?
 Statistical properties of the TM
 Transfer of information (image)

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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


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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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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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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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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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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Experimental Results :

Output Speckle (Eout)

Input Mask (Eobj)

Inversion Phase Conjugation Regularization
Reconstruction

C = 11% C = 76% C = 95%

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14/12/2011 32

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

From transmission matrix to reflection matrix

SLM
Linear sample CCD camera

= = =

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From transmission matrix to reflection matrix

SLM : array of pixels

Linear sample

CCD camera : array of pixels

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Reflection matrix and optical “DORT”


14/12/2011 36

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

Introduction

E0 KE0
Iterative time reversal

K * E0
*
KK * E0
*

K * KE0 KK * KE0

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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

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

Introduction

Works with an aberrating medium
(single scattering only)

Hypothesis: linearity, single scattering regime

14/12/2011 41

Setup

Scatterers:
100 nm isotropic gold
particles on a glass
slide

Cross Polarization
Control

Aberrating
medium

14/12/2011 42

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

Selective Focusing

Reflection

Control

14/12/2011 44

Setup

Scatterers:
100 nm isotropic gold
particles on a glass
slide

Cross Polarization

Aberrating
medium

14/12/2011 45

Adaptive optics

Aspect of the first input singular vector (phase mask)

Free space ~ lens With aberrating mediums

14/12/2011 46

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

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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14/12/2011 49


Experimental singular value distribution

? Pz dipole Px dipole
Number of SV

Py Dipole Pz Dipole
Px Dipole

14/12/2011 50

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

Complex envelope time reversal


14/12/2011 52

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

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

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

Setup
Setup

Modulation Part: Demodulation Part:
- 10 GHz arbitrary waveform generator - Interferometric detection of 2 quadratures
- Triple Mach-Zehnder modulator - 50 GHz oscilloscope

14/12/2011 56

Modulation / Demodulation

Modulation Part:
- 10 GHz arbitrary waveform generator
- Triple Mach-Zehnder modulator (Photline)
Demodulation Part:
- Interferometric detection of 2 quadratures
14/12/2011 - 20 GHz oscilloscope 57

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

Temporal focusing

Looped single
mode cavity
input output

Evanescent
coupling

Channel I Channel Q
Impulse
response

Numerical
time reversal
(correlations)
14/12/2011 59

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

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

Conclusion

- Spatial focusing
- Image transmission
- Singular value analysis
- Selective focusing through an aberrating medium
- Scattering pattern analysis
- Temporal focusing
- Towards spatial and temporal focusing...
14/12/2011 62

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

14/12/2011 63

Collaborateurs

Sylvain Geoffroy Alexandre Rémi Mathias Claude
GIGAN LEROSEY AUBRY CARMINATI FINK BOCCARA

14/12/2011 64


Hobs H. ref

Artefact :
« raster » effect
due to the
amplitude of Sref
Effect of ref

Observed Matrix

14/12/2011 65

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=
14/12/2011 80 25 μm 66

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
14/12/2011 67

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

14/12/2011 68

Applications : Focusing

Theoretical focusing
VS
Experimental focusing
Target Expected focusing from Experimental
measured matrix focusing

14/12/2011 69

Stability and Measurement Time

TM Measurement Time ~ 15 min
(1024x1024 )
Decorrelation Time of ~ 1 hour
ZnO deposit
Decorrelation Time of
<< 1s
Biological Tissues

14/12/2011 70

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

14/12/2011 71

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

14/12/2011 72

The matrix model : A conveniant model

Free space Multiply scattering sample

Detrimental to Conventional Optical Techniques
Matrix Description to link input / output k vectors

14/12/2011 73

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

14/12/2011 74

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)

14/12/2011 75

Transmission Matrix of an Optical Scattering Medium
Theoretical Focus Spot

D λF/D SLM L λl/L

l
F

14/12/2011 76

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

14/12/2011 77

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

14/12/2011 78

Spatio-temporal control of light in complex media

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