Scattering problems are concerned with how an inhomogeneous medium scatters an incident field. The direct scattering problem is to determine the scattered field from the incident field; the inverse scattering problem is to determine the nature of the inhomogeneity from the measured scattered field. These problems have played a fundamental role in diverse scientific areas such as radar and sonar, geophysical exploration, medical imaging, near-field and nano- optics. According to the Rayleigh criterion, there is a resolution limit to the sharpness of details that can be observed by conventional far-field imaging, one half the wavelength, referred to as the diffraction limit. It presents challenging mathematical and computational questions to solve the underlying inverse scattering problems with increased resolution due to the nonlinearity, ill-posedness, and large scale computation.

In this talk, our recent progress on a class of inverse surface scattering problems will be discussed. I will present new approaches to achieve subwavelength resolution for these inverse problems. Based on transformed field expansions, the methods convert the problems with complex scattering surfaces into successive sequences of two-point boundary value problems, where explicit reconstruction formulas are made possible. The methods require only a single incident field and are realized by using the fast Fourier transform. The convergence and error estimates of the solutions for the model equations will be addressed. I will also highlight some ongoing projects in rough and random surface imaging.