Transmission Matrix of an Optical Scattering Medium
Transcript of Transmission Matrix of an Optical Scattering Medium
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