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 ref E ref E 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 Control Spatial Light Modulator (SLM) in Phase Only Modulation A macropixel ↔ A k vector Output Detection CCD Camera A macropixel ↔ A k vector Transmission Matrix H Scattering sample Random Matrix Information is shuffled but not lost ! Output k Input k Statistical Properties of the TM Sebastien POPOFF Output k Free space Identity Matrix Information can be easily reconstructed Imaging, focusing… Input k ? Statistical properties uniform 2 out out E I 2 ref i out E e E I ref E 3 1 2 2 0 i out E I I iI eI 3 1 2 2 0 . i out ref I I iI eI 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 field 1..N obs ref in m m mn n n E E h E obs ref H H S diagonal Matrix representing the complex reference speckle Transmission Matrix Measured Matrix Amplitude of Reference Speckle induces correlation that modify the distribution ! We filter H obs to remove those correlations H fil obs fil mn mn obs mn m h h h « raster » effect due to the amplitude of S ref 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 t H H I H O A tradeoff : Tikhonov Regularization Initial speckle One point focusing Multiple point focusing ? * arg in t t et E HE Phase conjugated mask Resulting output pattern * arg . out t t et E HHE * t HH N=256 modes (16x16 pixels on the CCD) N=256 Expected focusing from measured matrix Experimental focusing Target Optimal Operator for σ = Noise variance Singular value distribution and fidelity of the reconstruction σ Reconstruction 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 obj E OE OH E We want OH close to Identity Inversion : 1 O H Perfect reconstruction Not stable in presence of noise OH I Very stable Reconstruction perturbated when the image is complex Phase Conjugation : * t O H We did : We can/will do : in n n mn out m E h E N .. 1 out E Output field in E 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 monochromatic transmission matrix in optics. It allows light focusing and detection through a complex medium. Having access to the transmission matrix opens the road to a better understanding of light transport. (Noiseless) 0 1 O H (Noisy) * t O H * H U V Tool : 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 i th channel. Σλ i 2 corresponds to the total transmittance for a plane wave