Abstract
From night vision and objects overwhelmed by sunlight to jammed signals and those that are purposely encrypted, detecting low-level or hidden signals is a fundamental problem in imaging. Here, we develop and exploit a new type of stochastic resonance, in which nonlinear coupling allows signals to grow at the expense of noise, to recover noise-hidden images propagating in a self-focusing medium. The growth rate is derived analytically by treating the signal–noise interaction as a photonic beam–plasma instability and matches experimentally measured resonances in coupling strength, noise statistics and modal content of the signal. This is the first observation of nonlinear intensity exchange between coherent and spatially incoherent light and the first demonstration of spatial coherence resonance for a dynamically evolving signal. The results suggest a general method of reconstructing images through seeded instability and confirm information limits predicted, but not yet observed, in nonlinear communications systems.
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References
Gammaitoni, L., Hanggi, P., Jung, P. & Marchesoni, F. Stochastic resonance. Rev. Mod. Phys. 70, 223–287 (1998).
Benzi, R., Parisi, G., Sutera, A. & Vulpiani, A. Stochastic resonance in climatic-change. Tellus 34, 10–16 (1982).
Nicolis, C. Stochastic aspects of climatic transitions—response to a periodic forcing. Tellus 34, 1–9 (1982).
Fauve, S. & Heslot, F. Stochastic resonance in a bistable system. Phys. Lett. A 97, 5–7 (1983).
Bezrukov, S. M. & Vodyanoy, I. Stochastic resonance in non-dynamical systems without response thresholds. Nature 385, 319–321 (1997).
Douglass, J. K., Wilkens, L., Pantazelou, E. & Moss, F. Noise enhancement of information-transfer in crayfish mechanoreceptors by stochastic resonance. Nature 365, 337–340 (1993).
Bulsara, A. R., Elston, T. C., Doering, C. R., Lowen, S. B. & Lindenberg, K. Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics. Phys. Rev. E 53, 3958–3969 (1996).
Simonotto, E. et al. Visual perception of stochastic resonance. Phys. Rev. Lett. 78, 1186–1189 (1997).
Vaudelle, F., Gazengel, J., Rivoire, G., Godivier, X. & Chapeau-Blondeau, F. Stochastic resonance and noise-enhanced transmission of spatial signals in optics: the case of scattering. J. Opt. Soc. Am. B 15, 2674–2680 (1998).
Blanchard, S., Rousseau, D., Gindre, D. & Chapeau-Blondeau, F. Constructive action of the speckle noise in a coherent imaging system. Opt. Lett. 32, 1983–1985 (2007).
Mitchell, M., Chen, Z. G., Shih, M. F. & Segev, M. Self-trapping of partially spatially incoherent light. Phys. Rev. Lett. 77, 490–493 (1996).
Christodoulides, D. N., Coskun, T. H., Mitchell, M. & Segev, M. Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett. 78, 646–649 (1997).
Shkunov, V. V. & Anderson, D. Z. Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media. Phys. Rev. Lett. 81, 2683–2686 (1998).
Sukhorukov, A. A. & Akhmediev, N. N. Coherent and incoherent contributions to multisoliton complexes. Phys. Rev. Lett. 83, 4736–4739 (1999).
Mendonca, J. T. & Tsintsadze, N. L. Analog of the Wigner–Moyal equation for the electromagnetic field. Phys. Rev. E 62, 4276–4282 (2000).
Soljacic, M., Segev, M., Coskun, T., Christodoulides, D. N. & Vishwanath, A. Modulation instability of incoherent beams in noninstantaneous nonlinear media. Phys. Rev. Lett. 84, 467–470 (2000).
Kip, D., Soljacic, M., Segev, M., Eugenieva, E. & Christodoulides, D. N. Modulation instability and pattern formation in spatially incoherent light beams. Science 290, 495–498 (2000).
Coskun, T. H., Grandpierre, A. G., Christodoulides, D. N. & Segev, M. Coherence enhancement of spatially incoherent light beams through soliton interactions. Opt. Lett. 25, 826–828 (2000).
Coskun, T. H. et al. Bright spatial solitons on a partially incoherent background. Phys. Rev. Lett. 84, 2374–2377 (2000).
Fedele, R. & Anderson, D. A quantum-like Landau damping of an electromagnetic wavepacket. J. Opt. B 2, 207–213 (2000).
Cohen, O. et al. Observation of random-phase lattice solitons. Nature 433, 500–503 (2005).
Dylov, D. V. & Fleischer, J. W. Observation of all-optical bump-on-tail instability. Phys. Rev. Lett. 100, 103903 (2008).
Dylov, D. V. & Fleischer, J. W. Multiple-stream instabilities and soliton turbulence in photonic plasma. Phys. Rev. A 78, 061804 (2008).
Mcnamara, B., Wiesenfeld, K. & Roy, R. Observation of stochastic resonance in a ring laser. Phys. Rev. Lett. 60, 2626–2629 (1988).
Martienssen, W. & Spiller, E. Coherence + fluctuations in light beams. Am. J. Phys. 32, 919–926 (1964).
Bulsara, A. R. & Gammaitoni, L. Tuning in to noise. Phys. Today 49, 39–45 (1996).
Mitra, P. P. & Stark, J. B. Nonlinear limits to the information capacity of optical fibre communications. Nature 411, 1027–1030 (2001).
Essiambre, R. J., Foschini, G. J., Kramer, G. & Winzer, P. J. Capacity limits of information transport in fiber-optic networks. Phys. Rev. Lett. 101, 163901 (2008).
Jia, S., Wan, W. & Fleischer, J. W. Forward four-wave mixing with defocusing nonlinearity. Opt. Lett. 32, 1668–1670 (2007).
Tsang, M., Psaltis, D. & Omenetto, F. G. Reverse propagation of femtosecond pulses in optical fibers. Opt. Lett. 28, 1873–1875 (2003).
Barsi, C., Wan, W. & Fleischer, J. W. Imaging through nonlinear media using digital holography. Nature Photon. 3, 211–215 (2009).
Weilnau, C. & Denz, C. Solitary beam formation with partially coherent light in an anisotropic photorefractive medium. J. Opt. A 5, S529–S535 (2003).
Vedenov, A. A. & Rudakov, L. I. Wave interaction in continuous media. Doklady Akademii Nauk Sssr 159, 767–770 (1964).
O'Neil, T. Collisionless damping of nonlinear plasma oscillations. Phys. Fluids 8, 2255–2262 (1965).
Chen, Z., Klinger, J. & Christodoulides, D. N. Induced modulation instability of partially spatially incoherent light with varying perturbation periods. Phys. Rev. E 66, 066601 (2002).
Sun, C., Dylov, D. V. & Fleischer, J. W. Nonlinear focusing and defocusing of partially-coherent spatial beams. Opt. Lett. 34, 3003–3005 (2009).
Cross, M. C. & Hohenberg, P. C. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).
Park, K., Lai, Y. C., Liu, Z. H. & Nachman, A. Aperiodic stochastic resonance and phase synchronization. Phys. Lett. A 326, 391–396 (2004).
Gilbreath, G. C. & Reintjes, J. F. Photorefractive Fourier image amplification for low-light-level image detection. Microwave Opt. Technol. Lett. 12, 119–123 (1996).
Heebner, J. E. & Boyd, R. W. Photorefractive optical recycling for contrast enhancement. Opt. Commun. 182, 243–247 (2000).
Breugnot, S., Rajbenbach, H., Defour, M. & Huignard, J. P. Low-noise photorefractive amplification and detection of very weak signal beams. Opt. Lett. 20, 447–449 (1995).
Shiratori, A. & Obara, M. Photorefractive coherence-gated interferometry. Rev. Sci. Instrum. 69, 3741–3745 (1998).
Carrillo, O., Santos, M. A., Garcia-Ojalvo, J. & Sancho, J. M. Spatial coherence resonance near pattern-forming instabilities. Europhys. Lett. 65, 452–458 (2004).
Freund, J. A., Schimansky-Geier, L. & Hanggi, P. Frequency and phase synchronization in stochastic systems. Chaos 13, 225–238 (2003).
Giacomelli, G., Giudici, M., Balle, S. & Tredicce, J. R. Experimental evidence of coherence resonance in an optical system. Phys. Rev. Lett. 84, 3298–3301 (2000).
Wu, B. B., Prucnal, P. R. & Narimanov, E. E. Secure transmission over an existing public WDM lightwave network. IEEE Photon. Technol. Lett. 18, 1870–1872 (2006).
Yang, Y. B., Jiang, Z. P., Xu, B. H. & Repperger, D. W. An investigation of two-dimensional parameter-induced stochastic resonance and applications in nonlinear image processing. J. Phys. A 42, 145207 (2009).
Deco, G. & Schurmann, B. Stochastic resonance in the mutual information between input and output spike trains of noisy central neurons. Physica D 117, 276–282 (1998).
Fraser, A. M. Reconstructing attractors from scalar time-series—a comparison of singular system and redundancy criteria. Physica D 34, 391–404 (1989).
García-Ojalvo, J. & Sancho, J. M. Noise in Spatially Extended Systems (Springer, 1999).
Acknowledgements
The authors would like to thank S. Verdú, E.E. Narimanov and P.R. Prucnal for valuable discussions. This work was supported by the National Science Foundation, the Department of Energy, and the Air Force Office of Scientific Research.
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Dylov, D., Fleischer, J. Nonlinear self-filtering of noisy images via dynamical stochastic resonance. Nature Photon 4, 323–328 (2010). https://doi.org/10.1038/nphoton.2010.31
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DOI: https://doi.org/10.1038/nphoton.2010.31
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