Image transmission through an opaque material

ARTICLE Received 26 May 2010 | Accepted 24 Aug 2010 | Published 21 Sep 2010 DOI: 10.1038/ncomms1078 Image transmission through an opaque material Sébastien Popoff1, Geoffroy Lerosey1, Mathias Fink1, Albert Claude Boccara1 & Sylvain Gigan1 Optical imaging relies on the ability to illuminate an object, collect and analyse the light it scatters or transmits. Propagation through complex media such as biological tissues was so far believed to degrade the attainable depth, as well as the resolution for imaging, because of multiple scattering. This is why such media are usually considered opaque. Recently, we demonstrated that it is possible to measure the complex mesoscopic optical transmission channels that allow light to traverse through such an opaque medium. Here, we show that we can optimally exploit those channels to coherently transmit and recover an arbitrary image with a high fidelity, independently of the complexity of the propagation. Institut Langevin, ESPCI ParisTech, CNRS UMR 7587, Universités Paris 6 & 7, INSERM, ESPCI, 10 rue Vauquelin, Paris, 75005, France. Correspondence and requests for materials should be addressed to S.G. (email: ). 1 nature communications | 1:81 | DOI: 10.1038/ncomms1078 | www.nature.com/naturecommunications © 2010 Macmillan Publishers Limited. All rights reserved.  ARTICLE nature communications | DOI: 10.1038/ncomms1078 I n a classical optical system, the propagation of a complex field from one plane to another is well understood, be it by Fresnel or Fraunhofer diffraction theory, or by ray tracing for more complex cases1. However, all these approaches break down when multiple scattering occurs2. A medium in which light is scattered many times mixes all input wavevectors in a seemingly random way, and is usually considered opaque. Until recently, scattering has always been considered as noise3, and most imaging techniques in turbid media rely on ballistic photons only4,5, which prevents the study of thick scattering samples. Following works in acoustics6, recent experiments have demonstrated that multiply scattered light can nonetheless be harnessed, thanks to wavefront control7–9, and even put to profit to surpass what one can achieve within a homo genous medium in terms of focusing10. In our experiment (see Fig. 1), we illuminate an object with a laser (displayed through a spatial light modulator (SLM)), and recover its image on a charge-coupled device (CCD) camera, after propagation through a thick opaque sample. As expected, we measure on the camera a speckle that bears no resemblance to the original image. This speckle is the result of multiple scattering and interferences in the sample. Although it can be described on average by the diffusion equation or Monte Carlo simulation11, the propagation through a real linear multiple scattering medium is too complex to be described by classical means. Nonetheless, multiple scattering is deterministic and informa tion is not lost. In other terms, the measured pattern on the CCD is the result of the transmission of light through a large number of very complicated optical channels, each of them with a given complex trans mission. Here, we study the inverse problem of the reconstruction of an arbitrary image, and show that it is possible to recover it through the opaque medium. A prerequisite is, however, to measure the so-called transmission matrix (TM) of our optical system. We define the mesoscopic TM of an optical system for a given wavelength as the matrix K of the complex coefficients kmn connect ing the optical field (in amplitude and phase) in the mth of M outputfree mode to the one in the nth of N input-free mode. Thus, the out projection Em of the outgoing optical field on the mth free mode is out given by Em = Σnkmn Enin where Enin is the complex amplitude of the optical field in the nth incoming free mode. In essence, the TM gives the relationship between input and output pixels, notwithstanding Reference part CC D Reconstruction P × NA 40 : 0. 75 Sample Controlled part × NA 20 :0 .5 P D Laser 532 nm L L P SLM Figure 1 | Experimental setup. A 532 nm laser is expanded and reflected off a spatial light modulator (SLM). The laser beam is phase-modulated, focused on the multiple scattering sample and the output intensity speckle pattern is imaged by a CCD camera. L, lens; P, polarizer; D, diaphragm. The object to image is synthetized directly by the SLM, and reconstructed from the complex output speckle, thanks to the transmission matrix.  the complexity of the propagation, as long as the medium is stable. A singular value decomposition of the TM gives the input and output eigenmodes of the system, and singular values are the amplitude trans mission of these modes. Inspired by various works in acoustics12,13 and electromagnet ism14, we demonstrated in Popoff et al.15 that it is possible to measure the TM of a linear optical system that comprises a multiple scat tering medium. In a nutshell, we send several different wavefronts with the SLM, record the results on the CCD and deduce the TM using phase-shifting interferometry. The singular value distribution of a TM of a homogeneous zone of the opaque sample follows the quarter-circle law (that is, there is no peculiar input/output correla tion16), which indicates that light propagation is in the multiple scat tering regime with virtually no ballistic photons left. Using this technique, we have access to K obs = K × Sref , where Sref is a diagonal matrix due to a static reference speckle. The input and output modes are the SLM and the CCD pixels, respectively. The measured matrix Kobs is sufficient to recover an input image. This TM measurement takes a few minutes, and the system is stationary well over this time. Once the matrix is measured, we generate an amplitude object Eobj by subtracting two-phase objects (see Methods for details). A realization takes a few hundred milliseconds, limited only by the speed of the SLM. Here, our aim is to use the TM to reconstruct an arbitrary image through the scattering sample: we need to estimate the initial input Eobj from the output amplitude speckle Eout. This problem consists in using an appropriate combination of the medium channels and, therefore, using a weighting of singular modes/singular values of the TM matched to the noise and to the transmitted image. Noises of different origins (laser fluctuations, CCD readout noise and residual amplitude modulation) degrade the fidelity of the TM measurement. It is the exact analogue of multiple-input multiple-output informa tion transmission in complex environment that has been studied in the past few years in wireless communications17. This inverse prob lem also bears some similarities to optical tomography18,19, although in a coherent regime20. We show that this allows us to reconstruct the image of an arbitrarily complex object, as viewed through an opaque medium. Results Reconstruction operators. There are two st (...truncated)