Brewster and pseudo-Brewster angles of uniaxial crystal surfaces and their use for determination of optical properties

Brewster and pseudo-Brewster angles are defined for surfaces of transparent and absorbing uniaxial crystals parallel and perpendicular to the optic axis. Two Brewster angles of a transparent uniaxial crystal surface parallel to the optic axis, measured when the optic axis is oriented perpendicular and parallel to the plane of incidence, readily determine the ordinary and extraordinary indices No and Ne. No and Ne can also be obtained from two Brewster angles measured on a surface perpendicular to the optic axis in contact with two media of different refractive indices. Conditions for the existence of two Brewster angles are discussed. The complex No and Ne of an absorbing uniaxial crystal can be derived from pseudo-Brewster-angle and minimum-reflectance data obtained in two symmetrical orientations of a surface parallel to the optic axis. An approximate, but accurate, explicit inversion procedure is presented for this purpose

Inversion of the nonlinear equations of reflection ellipsometry for uniaxial crystals in symmetrical orientations

The complex ordinary (No) and extraordinary (Ne) refractive indices of an absorbing uniaxial crystal can be determined using reflection ellipsometry. The measurements are taken with the optic axis parallel and perpendicular to the crystal’s surface. The equations obtained are solved without resort to iterative methods; No and Ne are determined separately. Sixteen solution sets (No, Ne) are obtained and the correct solution can be easily identified. We present an optimum angle of incidence that minimizes the relative errors in No and Ne

Combined reflection and transmission thin-film ellipsometry: a unified linear analysis

A scheme of combined reflection and transmission ellipsometry on light-transmitting ambient-film-substrate systems is proposed and the required sample design and instrument operation are investigated. A comparative study of the sensitivity of external and internal reflection and transmission ellipsometry is carried out based on unified linear approximations of the exact equations. These approximations are general in that an arbitrary initial film thickness is assumed. They are simple, because a complex sensitivity function is introduced whose real and imaginary projections determine the psi (Ψ) and delta (Δ) sensitivity factors. Among the conclusions of this paper are the following. (1) External reflection ellipsometry near the Brewster angle of a transparent ambient-substrate system is extremely sensitive to the presence of very thin interfacial films. For example, films as thin as 10-5 Å of gold are readily detectable on glass substrates at an angle of incidence 0.3° below the Brewster angle, assuming a measuring wavelength of 5461 Å with an ellipsometer of 0.05° precision. (2) The formation of thin nonabsorbing films at the interface between transparent ambient and substrate media is not detectable, to first order, as a change in the ellipsometric angle Ψ by either internal or external reflection or transmission ellipsometry. (3) The film-detection sensitivity of transmission ellipsometry increases monotonically with angle of incidence. (4) For each angle of external incidence there is a corresponding angle of internal incidence that leads to the same values of the reflection and transmission sensitivity functions. These angles are interrelated by Snell’s law. (5) The ranges of validity of the linear approximation in reflection and transmission ellipsometry are comparable. The case of total internal reflection ellipsometry may lead to strong nonlinear behavior of Ψ and Δ as functions of the film thickness in the range below 0.05 of the wavelength of light

Brewster and pseudo-Brewster angles of uniaxial crystal surfaces and their use for determination of optical properties

Brewster and pseudo-Brewster angles are defined for surfaces of transparent and absorbing uniaxial crystals parallel and perpendicular to the optic axis. Two Brewster angles of a transparent uniaxial crystal surface parallel to the optic axis, measured when the optic axis is oriented perpendicular and parallel to the plane of incidence, readily determine the ordinary and extraordinary indices No and Ne. No and Ne can also be obtained from two Brewster angles measured on a surface perpendicular to the optic axis in contact with two media of different refractive indices. Conditions for the existence of two Brewster angles are discussed. The complex No and Ne of an absorbing uniaxial crystal can be derived from pseudo-Brewster-angle and minimum-reflectance data obtained in two symmetrical orientations of a surface parallel to the optic axis. An approximate, but accurate, explicit inversion procedure is presented for this purpose