Abstract
Spectral properties of lamellar sub-wavelength gratings (nanogratings) are described by effective medium approximation (EMA). Analytical spectral formulae for ordinary and extraordinary effective optical functions are derived for nanogratings consisting of material described by Sellmeier, damped harmonic oscillator, and Drude formulae. Spectral origin for birefringence of dielectric nanogratings and linear dichroism for absorbing ones is discussed for model cases and gratings consisting of real natural materials. Simple approximation by zero-order diffraction is compared with rigorous modeling based on Rigorous Coupled Wave Analysis (RCWA). Limits of applicability of effective medium approximation is discussed in the spectral domain.
Highlights
Optical methods represented by reflection and transmission spectroscopy, ellipsometry, and polarimetry became nowadays standards of process monitoring and quality control in semiconductor technology
Drude model is in ellipsometric community understood as being basic model describing fundamental properties of metals and its optical function has the form: Peculiar behavior was found for the extraordinary effective optical function, that exhibits pure damped harmonic oscillator (DHO) behaviour
The spectral dependence of effective parameters of sub-wavelength lamellar grating from materials described by common optical function, namely Sellmeier formula, damped harmonic oscillator, and Drude formulae, was studied
Summary
Optical methods represented by reflection and transmission spectroscopy, ellipsometry, and polarimetry became nowadays standards of process monitoring and quality control in semiconductor technology. Modeling of optical properties of gratings and periodic systems is usually based on generalization of the Berreman matrix approach used for anisotropic multilayer systems [7]. One from the approaches to simplify modeling of the subwavelength gratings is to use the effective medium approximation (EMA), which replaces grating by an effective layer with new optical parameters. Effective parameters of isotropic lamellar gratings were intensively studied in the single wavelength context and optical constants of the effective layer in quasi-static limit (ratio between the grating period Λ and the incident light wavelength λ is close to zero) are very well known [10]. Spectral origin of interesting birefringent and dichroic properties of nanogratings is explained on example of three basic optical functions: Sellmeier relation describing dispersion in non-absorbing spectral range, damped harmonic oscillator describing absorption, and Drude model for free electron absorption in metals.
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