Accurately determining the optical constants (refractive index n and extinction coefficient k) of transparent solid materials is a crucial issue in optical design. The dual thickness transmittance method offers a straightforward approach that does not require the Kramers-Kronig relationship. For weakly absorbing materials, high-precision measurement results can be obtained by changing the thickness. The double thickness transmittance method establishes nonlinear equations about optical constants by measuring the transmittance of materials with two different thicknesses. Due to the complexity of the equations, it is difficult to obtain analytical solutions, and inversion methods are often used to solve optical constants. These inversion methods present challenges including computational time requirements, iterative errors, and multiple potential values for refractive index results. While researchers have attempted to address these issues through faster iterative algorithms or combinations with methods such as ellipsometry, the outcomes remain suboptimal.

Recent research has employed transmittance spectra of stacked samples combined with inversion methods to determine material optical constants, though the inherent limitations of inversion methods persist. An alternative approach utilizing dual spectral analysis for determining optical constants based on transmittance and reflectance spectra of single-layer samples has been proposed. However, this method faces challenges due to inconsistent experimental conditions between transmittance and reflectance measurements. The present study integrates the advantages of both stacked sample transmittance spectrum inversion and single-layer sample dual spectrum analysis methods. The approach involves measuring transmittance of various stacked sample combinations and deriving single-layer sample reflectance through algebraic operations. This enables optical constant determination without direct reflectance measurement while avoiding inversion method limitations. This analytical method based on stacked sample transmittance spectra significantly streamlines experimental measurement and calculation processes. For demonstration, transmittance measurements of Zinc Selenide samples under various stacking combinations were conducted using a Fourier transform infrared spectrometer in the 2?18 μm infrared band. The optical constants were determined using this novel method, followed by error analysis of extinction coefficient and refractive index measurements. Finally, the factors affecting the accuracy of optical constant measurement were studied by combining experiments and numerical simulations.

The optical constants of Zinc Selenide were measured using an application example. The relative uncertainty of the extinction coefficient k was less than 10% in the ranges of 3.5?5.5, 7.0?8.5, and 14?18 μm, and less than 5% in the range of 15.0?17.5 μm (Figs. 4 and 5). The relative uncertainty of refractive index n is less than 0.5% in the range of 3?15 μm, and less than 0.25% in the range of 7?12 μm (Figs. 6 and 7). Comparatively speaking, the measurement accuracy of refractive index is higher, and the extinction coefficient and its errors have almost no effect on the refractive index and its measurement accuracy. The main source of error for both is the measurement error of sample transmittance. The influence of sample thickness on the transmittance of stacked samples was studied through numerical simulation, and it was pointed out that sample thickness is crucial for reducing measurement errors in transmittance (Figs. 10 and 11). The new method proposes requirements for the transmittance t and thickness L of single-layer samples: under the condition of a spectrometer transmittance accuracy of 0.001, the transmittance t of single-layer sample should be greater than 0.004 and less than 0.953; adjusting the sample thickness L to ensure that the transmittance of the stacked sample falls as close as possible to the middle position between t² and t/(2-t), will help improve the measurement accuracy of transmittance and achieve accurate determination of optical constants.

This study combines the advantages of the stacked sample transmittance spectrum inversion method and the single-layer sample dual spectrum analysis method, and proposes an analytical model based on the stacked sample transmittance spectrum. It realizes the use of the single-layer sample dual spectrum analysis method to determine optical constants without measuring reflectance, while avoiding the disadvantages of inversion methods and greatly simplifying the experimental measurement and solution calculation process. The application example uses transmittance of Zinc Selenide stacked samples to demonstrate the specific use of the new method, and the results show that the method is feasible. Numerical simulation was conducted to study the influence of sample thickness on the transmittance of stacked samples. Adjusting the sample thickness reasonably to make the transmittance curve of stacked samples as close as possible to the middle position of its allowable range will help improve the measurement accuracy of transmittance and achieve accurate determination of optical constants. Our research results provide an alternative solution for the precise determination of the optical constants of transparent solids.