A Mathematical Theory of Super-Resolution by Using a System of Sub-Wavelength Helmholtz Resonators

Communications in Mathematical Physics

Volumn 337, Issue 1, 2015, Pages 379-428

  Ammari, Habib ^a   Zhang, Hai ^a  

^a ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84939950906     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-015-2301-4     Document Type: Article

References (42)

1. Abbe E.: Beiträge zür Theorie des Mikroskops und der mikroskopischen Wahrnehmung. Arkiv Mikroskopische Anat. 9, 413–468 (1873)
2. Ammari, H.: An Introduction to Mathematics of Emerging Biomedical Imaging. Math. and Appl., vol. 62. Springer, Berlin (2008)
3. Ammari, H., Bonnetier, E., Capdeboscq, Y.: Enhanced resolution in structured media. SIAM J. Appl. Math. 70, 1428–1452 (2009/10)
4. Ammari, H., Bretin, E., Garnier, J., Wahab, A.: Time reversal in attenuating acoustic media. In: Mathematical and Statistical Methods for Imaging. Contemp. Math., vol. 548, pp. 151–163. American Mathematical Society, Providence (2011)
5. Ammari, H., Garnier, J., Jing, W., Kang, H., Lim, M., Sølna, K., Wang, H.: Mathematical and Statistical Methods for Multistatic Imaging. Lecture Notes in Mathematics, vol. 2098, Springer, Cham (2013)
6. Ammari H., Kang H., Lee H.: Layer potential techniques in spectral analysis. In: Mathematical Surveys and Monographs, vol. 153. American Mathematical Society, Providence (2009)
7. Arsen’ev A.A.: The singularities of the analytic continuation and the resonance properties of the solution of a scattering problem for the Helmholtz equation. Dokl. Akad. Nauk SSSR 197, 511–512 (1971)
8. Bao G., Lin J.: Near-field imaging of the surface displacement on an infinite ground plane. Inverse Probl. Imaging 7, 377–396 (2013)
9. Bao G., Li P.: Near-field imaging of infinite rough surfaces. SIAM J. Appl. Math. 73, 2162–2187 (2013)
10. Bardos, C.: A mathematical and deterministic analysis of the time-reversal mirror. In: Inside Out: Inverse Problems and Applications. MSRI Publ., vol. 47. Cambridge Univ. Press, Cambridge (2003)
11. Bardos C., Fink M.: Mathematical foundations of the time reversal mirror. Asymptot. Anal. 29, 157–182 (2002)
12. Beale J.T.: Scattering frequencies of resonators. Commun. Pure Appl. Math. 26, 549–563 (1973)
13. Berry M.V.: A note on superoscillations associated with Bessel beams. J. Opt. 15, 044006 (2003)
14. Blomgren P., Papanicolaou G., Zhao H.K.: Super-resolution in time-reversal acoustics. J. Acoust. Soc. Am. 111, 230–248 (2002)
15. Burq N.: Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180, 1–29 (1998)
16. Candès E.J., Fernandez-Granda C.: Towards a mathematical theory of super-resolution. Commun. Pure Appl. Math. 67, 906–956 (2014)
17. Candès E.J., Fernandez-Granda C.: Super-resolution from noisy data. J. Fourier Anal. Appl. 19, 1229–1254 (2013)
18. Casse B.D.F., Lu W.T., Huang Y.J., Gultepe E., Menon L., Sridhar S.: Super-resolution imaging using a three-dimensional metamaterials nanolens. Appl. Phys. Lett. 96, 023114-1 (2010)
19. Chambers D.H., Berryman J.G.: Time-reversal analysis for scatterer characterization. Phys. Rev. Lett. 92, 023902-1 (2004)
20. Fernández-Suárez M., Ting A.Y.: Fluorescent probes for super-resolution imaging in living cells. Nat. Rev. 9, 929–943 (2008)
21. Fink M.: Time-reversal waves and super-resolution. J. Phys. Conf. Ser. 124, 012004 (2008)
22. Fink M.: Time reversed acoustics. Sci. Am. 281, 91–97 (1999)
23. Fink, M., Lemoult, F., de Rosny, J., Tourin, A., Lerosey, G.: Subwavelength focussing in metamaterials using far field time reversal. In: Acoustic Metamaterials. Springer Series in Materials Science, vol. 166, pp. 141–168 (2013)
24. Fouque J.-P., Garnier J., Papanicolaou G., Sølna K.: Wave Propagation and Time Reversal in Randomly Layered Media. Springer, New York (2007)
25. Gadyl’shin R.R.: Existence and asymptotics of poles with small imaginary part for the Helmholtz resonator. Russ. Math. Surv. 52, 1–72 (1997)
26. Gustafsson M.G.L.: Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy. J. Microsc. 198, 82–87 (2000)
27. Gustafsson M.G.L.: Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution. Proc. Nat. Acad. Sci. 102, 13081–13086 (2005)
28. Hell S.W.: Far-field optical nanoscopy. Science 316, 1153–1158 (2007)
29. Huang B., Babcock H., Zhuang X.: Breaking the diffraction barrier: super-resolution imaging of cells. Cell 143, 1047–1058 (2010)
30. Huang F.M., Zheludev N.I.: Super-resolution without evanescent waves. Nano Lett. 9, 1249–1254 (2009)
31. Landkof, N.S.: Foundations of Mordern Potential Theory, Die Grundlenhren der mathematischen Wissenschaften in Einzeldarstellungen Band, vol. 180. Springer, New York (1972)
32. Lemoult F., Fink M., Lerosey G.: Acoustic resonators for far-field control of sound on a subwavelength scale. Phys. Rev. Lett. 107, 064301 (2011)
33. Lemoult F., Lerosey G., de Rosny J., Fink M.: Time reversal in subwavelength-scaled resonant media: beating the diffraction limit. Int. J. Microw. Sci. Technol. 2011, 425710 (2011)
34. Lemoult F., Ourir A., de Rosny J., Tourin A., Fink M., Lerosey G.: Resonant metalenses for breaking the diffraction barrier. Phys. Rev. Lett. 104, 203901 (2010)
35. Lerosey G., de Rosny J., Tourin A., Fink M.: Focusing beyond the diffraction limit with far-field time reversal. Science 315, 1120–1122 (2007)
36. Mast T.D., Nachman A., Waag R.C.: Focusing and imaging using eigenfunctions of the scattering operator J. Acoust. Soc. Am. 102, 715–725 (1997)
37. Miles J.W.: Resonant response of harbours: an equivalent-circuit analysis. J. Fluid Mech. 46, 241–265 (1971)
38. Mosk A.P., Lagendijk A., Lerosey G., Fink M.: Controlling waves in space and time for imaging and focusing in complex media. Nat. Photonics 6, 283–292 (2012)
39. Rayleigh L.: The theory of the Helmholtz resonator. Proc. R. Soc. 92, 265–275 (1916)
40. Synge E.H.: A suggested method for extending microscopic resolution into the ultra-microscopic region. Philos. Mag. 6, 356–362 (1928)
41. Zhang X., Liu Z.: Superlenses to overcome the diffraction limit. Nat. Mater. 7, 435–441 (2008)
42. Zheludev, N.I.: What diffraction limit? Nat. Mater. 7, 420–422 (2008)