Transmission Matrix of an

Optical Scattering Medium

ESPCI-ParisTech

10 rue Vauquelin

75231 PARIS

France

Measure of the TM

No acces to phase

information !

Requires interfero-metric

stability for several minutes !

not uniform

OK as long as …..

…. is constant

refE

refE

Setup

Focusing Image Detection

Objective : Measuring the Transmission Matrix

Hypothesis : Coherence of the illumination, Stability of the Medium, Linearity

Mathias FINK

Claude BOCCARA

Geoffroy LEROSEY

Sylvain GIGAN

Input ControlSpatial Light Modulator (SLM)

in Phase Only Modulation

A macropixel ↔ A k vector

Output DetectionCCD Camera

A macropixel ↔ A k vector

Transmission Matrix H

Scattering

sample

Random Matrix

Information is

shuffled but not lost !

Outp

ut

k

Input k

Statistical Properties of the TM

Sebastien POPOFF

Outp

ut

k

Free space

Identity Matrix

Information can be

easily reconstructed

Imaging, focusing…

Input k

?

Statistical properties

uniform

2

outout EI

2

ref

i

out EeEI

refE3 1

2 20 i

outE I I i I e I

3 1

2 20 .i

out refI I i I e I E E

Measure of the Amplitude of the Field

Construction of the Transmission Matrix

Principle : For each component of the input basis we measure the resulting output field1..N

obs ref in

m m mn n

n

E E h E obs refH H S diagonal Matrix representing the

complex reference speckle

Transmission MatrixMeasured Matrix

Amplitude of Reference Speckle induces correlation that modify the distribution !

We filter Hobs to remove those correlations Hfil

obsfil mn

mn obs

mnm

hh

h

« raster » effectdue to the

amplitude of Sref

Observed Matrix Filtered Matrix

« Quarter-circle law » predicted by Random Matrix Theory(V. Marcenko and L. Pastur, Sbornik : Mathematics, 1967)

Sample

Deposit of ZnO

L = 80 25 μm

l* = 6 2 μm

1* *.

t tH H I HOA tradeoff : Tikhonov Regularization

Initial speckle One point focusing Multiple point focusing

?* argin t t etE H EPhase conjugated mask

Resulting output pattern* arg.out t t etE H H E

*tH H

N=256 modes (16x16 pixels on the CCD)

N=2

56

Expected focusing frommeasured matrix Experimental focusing

Target

Optimal Operator for

σ = Noise variance

Singular value distribution and fidelity of the reconstruction

σ

Re

co

ns

tru

cti

on

Input Mask (Eobj)

Output Speckle (Eout)

Inversion Phase Conjugation Regularization

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

Conclusion and Perspectives

- transfered information through complex medium (Focusing, Imaging) Develop a faster setup (micromirror arrays, ferromagnetic SLMs) for biological purposes

- studied statistical properties of a scattering medium Study more complex media (Anderson localization, photonic christals, Levy glasses…)

Some focusing experiments (full resolution)

Comparing experimental and expected focusing for one focus spot

(A.N.Tikhonov, Soviet. Math. Dokl., 1963)

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, Nature Communications, http://arxiv.org/abs/1005.0532

What operator to reconstruct a complex image ?

Which phase mask to apply to focus through the medium ? . .img out objE O E OH E We want OH close to Identity

Inversion :1O H Perfect reconstruction

Not stable in presence of noiseOH I

Very stable

Reconstruction perturbated when the

image is complex

Phase Conjugation : *tO H

We did : We can/will do :

in

n

n

mn

out

m EhEN..1

outE Output field

inE Input field

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, Nature Photonics 2, 110 (2008).

We experimentally measure and study the monochromatictransmission matrix in optics. It allows light focusing and detectionthrough a complex medium. Having access to the transmission matrixopens the road to a better understanding of light transport.

(Noiseless)01O H (Noisy)

*tO H

*H U VTool : Singular Value Decomposition

Output basis

Input basis

We study the distribution of (normalized) singular values ρ(λ)

1

2

0 0 0

0 0 0

0 0 ... ...

0 0 ... N

i >0 represents the energy transmission through the ith channel.

Σλi2 corresponds to the total transmittance for a plane wave