Abstract: An apparatus, system and method are described for determining profiles of the absolute or relative concentrations of N constituent chemical components of a chemical mixture over a group of samples. For each sample a plurality of sets of concentrations is determined wherein each set has substantially the same error for corresponding to the absorption spectra measured for the composite sample. A plurality of samples of the chemical mixture are generated, for example, by removing parts of a surface sequentially. From amongst the plurality of sets of absolute or relative concentrations over the plurality of samples, a path of highest probability is determined from which profiles of each of the chemical components can be determined. The invention is applied to generating profiles of chemical constituents as material is received from a surface of a semiconductor chip to generate profiles as a function of depth into the sample.

Abstract: An apparatus and method are described for determining the relative concentrations of the N constituent chemical components of a chemical combination. The spectral response, C.sub.i, of the composite sample is measured at M wavelengths .lambda..sub.i, wherein M.gtoreq.N. The relative concentration of the jth constituent is X.sub.j. The spectral response of the jth component at wavelength .lambda..sub.i is A.sub.ij. the set of M equations ##EQU1## have intersections defining values of X.sub.j of which only those within the region defined by 0.ltoreq.X.sub.j are possible values of X.sub.j. An error is assigned to each intersections and the intersection of minimum error defines the values of X.sub.j. Such a determination is rapid. A profile of each of the relative concentration X.sub.j in the sample is readily and rapidly determined.

Abstract: An apparatus and method are described for determining the relative concentrations of the constituent chemical components of a chemical combination. The spectral response of the composite sample, measured at a number of wavelengths .lambda..sub.i, is represented by a vector C. The relative concentration of the jth constituent is X.sub.j, and the relative concentrations of each of the chemical constituents are represented by the vector X. The vector X can be determined from the vector C by the following matrix equation:X=(A.sup.T A).sup.-1 A.sup.T Cwherein the matrix A has elements A.sub.ij, which is the spectral response of the jth component at wavelength .lambda..sub.i. Since the matrix A is a predetermined set of numbers, the elements of vector X are readily determined from the above equation. Such a determination is rapid; a profile of each of the constituents X.sub.j in the sample is readily and rapidly determined.