Near-field optics is an emerging research topic in the past fewdecades, mostly motivated by applications in the near-fieldmicroscopy in an effort to break the diffraction limit. In thefar-field imaging, only the propagating wave components with spatialfrequency below the wavenumber are available, and it is well-knownthat the resolution of the image is approximately half of the wavelength(diffraction limit). In the near field, however, the bandwidth ofthe spatial frequency may be expanded by taking account of theevanescent (exponentially decayed) waves. Nowadays there existvarious configurations for the near-field microscopies. However, itis recognized that the images obtained from the near-fieldmicroscopies are problematic by visualizing the object in ananalogical way. Therefore, the inverse scattering theory is appliedto understand how the structure of the scattering object is encodedin the measured scattered field. When single scattering (or Bornapproximation) is assumed, the studies are complete for thenear-field scanning optical microscopy (NSOM) and the total internalreflection microscopy (TIRT) within the framework of the inversescattering theory.In this thesis, we focus on one specific problem where the imagingtarget is a ground plane with some local disturbance. The data iscollected in the near-field regime with a distance above the surfacedisplacement that is smaller than the wavelength. In the recentpaper by G. Derveaux, G. Papanicolaou, and C. Tsogka, a linearizedmodel has been introduced for the nonlinear inverse scatteringproblem by the single scattering assumption. The authors alsoproposed a broadband imaging strategy for denoising and improvingthe resolution of the image. In the thesis, we investigate the moregeneral case by considering the full scattering model, for which thelinearized model is no longer valid. By the analysis of thescattered field, it is confirmed that the evanescent wave modeswhich are not accessible in the far-field regime become significantin the near field. Evanescent wave modes make it possible to breakthe diffraction limit. It is shown that such exponentially decayedmodes of the scattered wave contain the high spatial frequencyinformation (fine features) of the profile. We formulate explicitlythe connection between the evanescent wave modes and the highfrequency components of the surface displacement, and present a newnumerical scheme to reconstruct the surface displacement from theboundary measurements. By extracting the information carried by theevanescent modes effectively, it is shown that the resolution of thereconstructed image is significantly improved in the near field.Numerical examples show that images with a resolution oftenth of wavelength are obtained.To overcome the ill-posedness and the presence of local minimaassociated with this nonlinear imaging problem, we propose to usemultiple frequency data to image the profile of the surfacedisplacement in the second part of the thesis. The main idea is tomarch from the lowest wavenumber to the highest wavenumber. At thefixed wavenumber, by an analysis of the domain derivative for theforward scattering map, a vector field is chosen such that thedefined cost functional decreases. The reconstructed profile evolveswith the chosen vector field at the fixed wavenumber and theevolution process continues until it reaches the highest wavenumber.The proposed reconstructed scheme is able to capture the mainfeature of the profile at low frequency and recover the fine detailsat higher frequency. In particular, for a multiple scale profile, itresolves the fine scale with sufficiently high frequency.
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