Patent ID: 7212933

Claim:
A method for analyzing inverse scattering spectral components comprising the steps of: irradiating an object with a measuring wave; measuring a reflection spectrum of the object; measuring a transmission spectrum of the object; calculating a transmission coefficient on a computer from: t k = 1 - ik 2 ⁢ ∫ - ∞ + ∞ ⁢ ⁢ ⅆ ze ikz ⁢ V ⁡ ( z ) ⁢ ψ k + ⁡ ( z ) , where V(z) is the location interaction between the object and ψ k + (z) is the measuring wave, calculating a reflection coefficient on the computer from: r k = - ik 2 ⁢ ∫ - ∞ - ∞ ⁢ e - ikz ⁢ V ⁡ ( z ) ⁢ ψ k + ⁡ ( z ) using a set of definitions t k ⁢ ψ ~ k ⁡ ( z ) = ψ k + ⁡ ( z ) r k t k = r ~ k V ~ 1 ⁡ ( z ) = ∫ - ∞ + ∞ ⁢ ⁢ ⅆ ( 2 ⁢ ⁢ k ) ⁢ 2 ⁢ ⁢ i k ⁢ r ~ k ⁢ e - 2 ⁢ ikz to convert a Lippmann-Schwinger inverse scattering equation ψ k + ⁡ ( z ) = e ikz - ik 2 ⁢ ∫ - ∞ + ∞ ⁢ ⁢ ⅆ z ′ ⁢ e ik ⁢  z - z ′  ⁢ V ⁡ ( z ′ ) ⁢ ψ k + ⁡ ( z ′ ) on the computer in a Volterra-type form V ~ 1 ⁡ ( z ) = ∫ - ∞ + ∞ ⁢ ⁢ ⅆ ( 2 ⁢ k ) ⁢ ⅇ - 2 ⁢ ⁢ ⅈ ⁢ ⁢ kz ⁢ 2 ⁢ i k ⁢ r k ⁡ [ 1 + i ⁢ ⁢ k ⁢ ⁢ Δ 2 ⁢ ∑ j ⁢ ⁢ ⅇ - ⅈ ⁢ ⁢ k ⁢ ⁢ z j ⁢ V ⁡ ( z j ) ⁢ ψ ~ k ⁡ ( z ) ] ; and iterating the Volterra-form of the Lippmann-Schwinger equation on the computer to produce an approximate solution {tilde over (V)} 1 (z), where {tilde over (V)} 1 (z) is absolutely and uniformly convergent.