Depth profiling the optical absorption and thermal reflection coefficient via an analysis based on the method of images (abstract)

Abstract

The problem of depth profiling optical absorption in a thermally depth variable solid is a problem of direct interest for the analysis of complex structured materials. In this work, we introduce a new algorithm to solve this problem in a planar layered sample which is impulse irradiated. The sample is comprised of “N” model layers of thickness Δx, of constant diffusivity α, where the conductivity varies depth wise with each layer. This derivation extends to the general case of a depth variable thermal reflection coefficient with depth variable optical source density. In such a sample, at finite time, t, past excitation, thermal energy can only significantly penetrate NL model layers NL≈4αt[−ln(ε)]/2Δx, where ε is a small error (ε⩽10−6) and a double transit through each layer is assumed. The depth profile of optical absorption in each layer, i, is approximated by δ(x−iΔx), weighted by the optical source density Si. The temperature at x=0− just inside a front medium contacting the sample is given by T(x=0,t)= ∑ i=12NLSi⋅GR(x,x0=iΔx,t)]x=0, where GR(x,x0,t) represents an effective Green’s function for optical absorption at the depth x0=iΔx in the sample. The method of images1 gives GR(x,x0=iΔx,t) in the following form: [GR(x,0Δx,t)GR(x,2Δx,t)…GR(x,2NLΔx,t)]=[A10 A12 A14 A16 …..A1,2NL0 A32 A34 A36 …..A3,2NL….0……A2NL−1,2NL][G(x−0Δx,t)G(x−2Δx,t)……G(x−2NLΔx,t)]. The G(x−nΔx,t) are shifted image fields obtained from the infinite domain Green’s function for one-dimensional heat conduction. They account for thermal wave reflection/transmission over the path length nΔx from the source (at interface i) to the surface (x=0). The Ain are lumped coefficients giving the efficiency of heat transmission from the ith source to the surface for each path order n. They are determined by a mapping procedure that identifies all propagation paths of each order, n, and computes the individual and lumped reflection coefficients. Equation (2) is written for sources placed at odd ordered model interfaces. A similar upper triangular matrix results for the placement of sources at the even ordered interfaces. Recovery of the optical absorption profile proceeds by inversion of Eq. (1) for known GR(x,iΔx,t). Determination of the kernel requires a solution of the related type II inverse problem.2,3 An evaluation of this procedure and its conditioning will be presented.