Spectrophotometry of Layers on Plane Parallel Substrates
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ICAL OPTICS
Spectrophotometry of Layers on Plane Parallel Substrates A. B. Sotskya, *, S. S. Mikheeva, N. I. Stas’kova, and L. I. Sotskayab a
Mogilev State A. Kuleshov University, Mogilev, 212022 Belarus b Belarusian‒Russian University, Mogilev, 212000 Belarus *e-mail: [email protected]
Received March 5, 2020; revised March 5, 2020; accepted March 28, 2020
Abstract—Integral expressions are obtained for the reflectance and transmittance spectra of a structure that is formed by two thin layers deposited on opposite faces of a plane-parallel substrate and that is obliquely illuminated with partially coherent light. As a result of an asymptotic analysis of the integrals, approximate analytical formulas are found for calculating the indicated spectra, which are convenient for use to solve inverse spectrophotometry problems. An aluminum doped zinc oxide layer deposited on a glass substrate is studied. The spectra of the refractive indices and absorbances of the layer and the substrate, as well as the layer thickness, are restored by processing the reflectance and transmittance spectra of the structure measured for s- and p-polarized waves at two angles of incidence of the light on the structure. The found parameters of the structure are used in computational experiments to estimate the applicability limits of the formulated approximations. Keywords: spectrophotometry, partial coherence, inverse optical problem, reflectance, transmittance DOI: 10.1134/S0030400X20080354
INTRODUCTION To monitor the geometric and optical characteristics of layers of various materials, the spectrophotometry method has been widely used. The majority of spectrophotometers are characterized by the use of broadband light sources with separation of a narrow wavelength interval
λ 0 − Δλ ≤ λ ≤ λ 0 + Δλ
(Δλ /λ 0 1)
(1)
using a tunable monochromator. In this situation, the radiation that probes a layer is partially coherent. This specific feature can significantly affect the measured spectra of reflectance R(λ) and transmittance T(λ) of the examined structure. In particular, partial radiation coherence should be taken into account upon solving inverse spectrophotometry problems for layers located on plane-parallel weakly absorbing substrates [1–12]. In works [1, 2, 5, 7, 9–11], various versions of the phenomenological approach were used, in which the so-called coherent and incoherent homogeneous layers are distinguished in the plane-parallel structure under study [9]. For coherent layers whose thickness is comparable to λ, the reflection and transmission coefficients of light are calculated in terms of the monochromatic light model. A plane-parallel substrate with refractive index ns and thickness D λ 2(2ns Δλ)−1 [7] is assumed to be an incoherent layer. This layer is treated in the ray approximation, in which rays of different orders reflected from its boundaries are consid-
ered to be incoherent. The resulting expressions for R(λ) and T(λ) are obtained by summation of Airy series. In the explicit form, these expressions are rather cumber
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