Backpropagation Imaging in Nonlinear Harmonic Holography in the Presence of Measurement and Medium Noises

Abstract

In this paper, the detection of a small reflector in a randomly heterogenous medium using second-harmonic generation is investigated. The medium is illuminated by a time-harmonic plane wave at frequency ω. It is assumed that the reflector has a non-zero second-order nonlinear susceptibility, and thus emits a wave at frequency 2ω in addition to the fundamental frequency linear scattering. It is shown how the fundamental frequency signal and the second-harmonic signal propagate in the medium. A statistical study of the images obtained by migrating the boundary data is performed. It is proved that the second-harmonic image is more stable with respect to medium noise than the one obtained with the fundamental signal. Moreover, the signal-to-noise ratio for the second-harmonic image does not depend neither on the second-order susceptibility tensor nor on the volume of the particle. Mathematics Subject Classification (MSC2000): 35R30, 35B30

Surface Plasmon Resonance of Nanoparticles and Applications in Imaging

“…The scalar case was recently considered in [13]. Our approach in this paper, combined with the ones in [7,8], opens also a door for a numerical and mathematical framework for optimal shape design of resonant nanoparticles and their superresolved imaging. The limit when t → 0 − is computed similarly and we find (4.8) for k = 0.…”

A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers

“…k,1 and σ (1) k,2 are the linear scattering coefficients for the k-th point scatterer, σ (2) k,1 and σ (2) k,2 are the quadratically nonlinear scattering coefficients for the k-th point scatterer, and φ (j) k is the external field acting on the k-th point scatterer at frequency ω j . The fields φ (j) satisfy the following relationship:…”

“…We consider the imaging of extended scatterers with two quadratically nonlinear point scatterers. The nonlinear scattering coefficients are σ (1) k,1 = σ (1) k,2 = 0.5 and σ (2) k,1 = σ (2) k,2 = 0.4. The numerical performance is shown in Table 4 for the following two examples.…”

“…(r, ω 1 , ω 2 , ω 3 ) = χ (2) (r, ω p(1) , ω p(2) , ω p(3) ), where {p(1), p(2), p(3)} is a permutation of {1, 2, 3}. Suppose that a source of frequency ω is incident upon a cubic nonlinear medium.…”

“…generically take different values, so do σ(2) k,1 and σ(2) k,2 . (iii) For a collection of cubically nonlinear point scatterers located at r k ∈ R 2 , k = 1, .…”