Birefringence Analysis of a Uniaxially Anisotropic Substrate Based on the Trajectory of the Transmission Ellipsometric Pseudoconstant in Polar Coordinates

유사 투과타원상수의 극좌표상 자취에 기반한 단축 이방성 기층의 복굴절 해석

Abstract

The trajectory of the transmission ellipsometric pseudoconstant ${\rho}=tan{\psi}_{\mu}e^{i{\Delta}_{\mu}}$ of a uniaxially anisotropic substrate like PET forms a circle in polar coordinates, as the phase-retardation angle is varied at a fixed azimuthal angle. The radius as well as the center's position of this circle are functions of the azimuthal angle only. This circle passes through the point (1,0), and the center of this circle is located on the real axis. These characteristics of the circle are examined analytically, and are utilized to derive simple expressions for the azimuthal angle and the phase-retardation angle of the uniaxially anisotropic substrate using the measured transmission ellipsometric constant. Finally, we confirm that the derived expressions are well applied to the analysis of the optical anisotropy of a PET film.

PET와 같은 단축 이방성 기층의 유사 투과타원상수 ${\rho}=tan{\psi}_{\mu}e^{i{\Delta}_{\mu}}$는 광축의 방위각이 고정되어 있을 때 위상지연각이 변함에 따라 극좌표계에서 원을 그린다. 이 원의 반지름과 중심의 좌표는 광축의 방위각에 의해 결정되지만 항상 좌표점 (1,0)을 지나며 원의 중심은 실수축 상에 있다. 단축 이방성 기층의 유사 투과타원상수가 극좌표에서 그리는 자취의 이러한 특성들을 해석적으로 확인하고 이 특성들을 이용하여 측정된 투과타원상수로부터 단축 이방성 기층의 광축의 방위각과 위상지연각을 결정하는 간단한 표현들을 제시하고 실제 PET 시료의 이방성 분석에 잘 적용됨을 보였다.

Fig. 1. Graph of 1/ (1 – cot2θ) vs θ.

Fig. 2. Graph of 1/ (1 – tan2θ) vs θ.

Fig. 3. Ellipsometric pseudoconstant in polar coordinates when tan$\psi$_μ < tanP, and a representative circle.

Fig. 4. Ellipsometric pseudoconstant in polar coordinates when tan$\psi$_μcosΔ_μ > tanP, and a representative circle.

Fig. 5. Ellipsometric pseudoconstant in polar coordinates when tan$\psi$_μcosΔ_μ < tanP and tan$\psi$_μ > tanP, and a representative circle.

Fig. 6. Measured transmission ellipsometric spectra (open symbols) of a PET film in polar coordinates at a few selected azimuthal angles with the best fit calculated spectra (solid lines).

Fig. 7. Distance of the measured ellipsometric pseudoconstant ρ from the point (1,0) in polar coordinates.

Fig. 8. Calculated phase retardation angles using the measured ellipsometric pseudoconstant show a well-defined normal dispersion. The retardation angles from 5 different sample azimuthal angles overlap each other very well.

Fig. 9. Variation of the birefringence dispersion Δn as the order N in Eq. (26) is increased from 0 to 3.

Fig. 10. Dispersion of the absolute phase retardation of the PET film analyzed in this research (open circles) matches perfectly with the previously obtained one (line) using the conventional transmission spectroscopic ellipsometry.