pseudo-Brewster Angle Research Articles

Angle scanning reflectometry: study of two characteristic isoreflectance angles

On analysing transparent thin films by angularly scanning light passing through a prism, one finds two characteristic angles theta A and theta B where the reflectance is equal for layers of uniform thickness. The first angle is found for a particular sequence of thicknesses and valid for both p- and s-polarization. The second is the well known pseudo-Brewster angle, corresponding to a null Fresnel reflection coefficient at the film-air interface and is valid for p-polarization only. These two angles allow the determination of the dielectric constant and the thickness of the layer. The use of these angles is also discussed in the case of weakly absorbing films and for metals when covered by dielectric overlayers.

Brewster and Pseudo-Brewster Angle Technique for Determination of Optical Constants

The analytic solution of the pseudo-Brewster angle, for the case of electromagnetic waves incident from a transparent optical material onto an absorbing material, has been obtained. We have developed a unique measurement method, called the pseudo-Brewster angle method, for obtaining the optical constants of thin films. By this method, the reflected intensity is measured at any angle of incidence. The optical constants are calculated from the pseudo-Brewster angle at the minimum value of the reflection coefficient. As a result, the pseudo-Brewster angle method is very useful for measuring the optical constants of a thin film.

Optical properties of cooled Rhodamine B in ethanol

The optical properties of an organic laser dye luminophor, Rhodamine B in ethanol, were investigated at both high and low concentrations as a function of temperature. At high concentrations the absorption spectra (extinction coefficient, k) and the refractive index (n) were determined by measuring the reflectivity of parallel-polarized light at the pseudo-Brewster angle at the dye–air interface. The absorption spectra at low concentration were also measured and compared with those at high concentration. It was found that at high concentration, independent of temperature, the dye–dye interaction among neighboring dye molecules contributes to the deviation of the absorption spectra from Beer’s law. The dependence of absorption spectral distribution on concentration becomes more pronounced as the solvent temperature is reduced. This is thought to be mainly due to the mutual interaction of neighboring molecules and also to the formation of higher-order aggregates. At low concentrations, as the temperature is reduced, the absorption spectra show little change and, in contrast to those at high concentration, obey the Beer–Lambert law; i.e., the absorption cross sections are independent of concentration. The absorption intensity of concentrated solutions decreases markedly compared with those at low concentrations.

Second condition for the pseudo-Brewster angle of highly absorbing media

An approximate but accurate second condition for the pseudo-Brewster angle ϕPB, valid in a range of n and k values, that has the advantage of replacing the well-known cubic equation for cot2ϕPB by a quadratic equation for sin2ϕPB is derived. It is also found that the optical constants of most semiconductors in the visible and near-ultraviolet ranges lie within the region of applicability of this second pseudo-Brewster-angle condition.

Transverse magnetic loss of metallic whispering gallery waveguides at 106 μm

The TM losses of metallic whispering gallery waveguides in the IR have been calculated for a range of angles of incidence. The allowed angles of incidence have been calculated from the caustic of the rays associated with particular modes. In many practical situations the pseudo-Brewster angle is large compared with the allowed ray angles, so that the usual assumptions must be reconsidered. In that case we find that the TM losses are much lower than generally assumed and that the TM and TE loss ratio is independent of the wall material. Experimental verification of these conclusions is presented.

Internal Reflection in Uniaxial Crystals, III: Transmission and Reflection Coefficients for an Extraordinary Incident Wave

Abstract The boundary conditions have been solved and the reflection and transmission coefficients calculated for an interface between a uniaxial crystal and an isotropic medium when the incident wave is extraordinary. The existence of a pseudo-Brewster angle for each reflected wave has been verified, and the polarizations of the reflected refracted waves have been determined.

Internal Reflection in Uniaxial Crystals II: Coefficients of Transmission and Reflection for an Ordinary Incident Wave

Abstract The boundary condition at the interface between a crystal and an isotropic medium have been solved in a general form for the situation when an ordinary ray is incident from the crystal and gives rise to two reflected rays besides the refracted ray. With the formulae obtained, the reflection and transmission coefficients have been calculated as functions of the incidence angle, and a pseudo-Brewster angle was observed to exist for each one of the reflected rays.

Contours of constant pseudo-Brewster angle in the complex ∊ plane and an analytical method for the determination of optical constants

The locus of all points in the complex plane of the dielectric function ?[?(r) + j?(i) = |?| exp(jtheta)], that represent all possible interfaces characterized by the same pseudo-Brewster angle theta(p)B of minimum p reflectance, is derived in the polar form: |?| = l cos(zeta/3), where l = 2(tan(2)Phi(p)B)k, zeta = arccos(- costheta cos(2)Phi(p)B/k(3)), and k = (1 - 2/3 sin(2)Phi(p)B)(1/2). Families of iso-Phi(p)B contours for (I) 0 degrees </= Phi(p)B </= 45 degrees and (II) 45 degrees </= Phi(p)B </= 75 degrees are presented. In range I, an iso-Phi(p)B contour resembles a cardioid. In range II, the contour gradually transforms toward a circle centered on the origin as Phi(p)B increases. However, the deviation from a circle is still substantial. Only near grazing incidence (Phi(p)B > 80 degrees ) is the iso-Phi(p)B contour accurately approximated as a circle. We find that |?| < 1 for Phi(p)B < 37.23 degrees , and |?| > 1 for Phi(p)B > 45 degrees . The optical constants n,k (where n + jk = ?((1/2)) is the complex refractive index) are determined from the normal incidence reflectance R(0) and Phi(p)B graphically and analytically. Nomograms that consist of iso-R(0) and iso-Phi(p)B families of contours in the nk plane are presented. Equations that permit the reader to produce his own version of the same nomogram are also given. Valid multiple solutions (n,k) for a given measurement set (R(0),phi(p)B) are possible in the domain of fractional optical constants. An analytical solution of the (R(0),Phi(p)B) ? (n,k) inversion problem is developed that involves an exact (noniterative) solution of a quartic equation in |?|. Finally, a graphic representation is developed for the determination of complex ? from two pseudo-Brewster angles measured in two different media of incidence.

Optical Surface Methods for Detection of Nucleic Acid Binding

Abstract Detection of DNA-binding and hybridization using pseudo-Brewster angle reflectometry and ellipsometry is described. DNA was attached to planar silicon surfaces coated with a 150 Å layer of nitrocellulose or with a monolayer of aminosilane. Poly-L-lysine was used as a linker between the modified surfaces and the DNA. The course of the reassociation was monitored continuously with the reflectometry method. A reassociation time of approximately one hour was observed for polynucleotides of 0.5–1.0 kb.

Extrema of the magnitude and the phase of a complex function of a real variable: application to attenuated internal reflection

Given a complex function F(ω) = |F(ω)|exp[jΔ(ω)] of a real argument ω, the extrema of its magnitude |F(ω)| and its phase Δ(ω), as functions of ω, are determined simultaneously by finding the roots of one common equation, Im[G(ω)] = 0, where G = (F′/F)2 and F′ = ∂F/∂ω. The extrema of |F| and Δ are associated with Re G 0, respectively. This easy-to-prove theorem has a wide range of applications in physical optics. We consider attenuated internal reflection (AIR) as an example. In AIR the complex reflection coefficient for the p polarization, rp(ϕ), and the ratio of complex reflection coefficients for the p and s polarizations, ρ(ϕ) = rp(ϕ)/rs(ϕ), are considered as functions of the angle of incidence ϕ. It is found that the same (cubic) equation that determines the pseudo-Brewster angle of minimum |rp| also determines a new angle at which the reflection phase shift δp = arg rpexhibits a minimum of its own. Likewise, the same (quartic) equation that determines the second Brewster angle of minimum |ρ| also determines angles of incidence at which the differential reflection phase shift Δ = arg ρ experiences a minimum and a maximum. Angular positions and magnitudes of all extrema are calculated exactly for a specific case that represents light reflection by the vacuum–Al or glass–aqueous-dye-solution interface. As another example, the normal-incidence reflection of light by a birefringent film on an absorbing substrate is examined. The ratio of complex principal reflection coefficients is considered as a function of the film thickness normalized to the wavelength of light. The absolute value and the phase of this function exhibit multiple extrema, the first 13 of which are determined for a specific birefringent film on a Si substrate.

Infrared reflectivity of metallic surfaces

We propose simple and good approximate equations for the reflectance of the perpendicular and parallel polarizations of light upon an isotropic absorbing medium deduced from the Fresnel coefficients of reflection. We also calculate the principal incidence and the pseudo-Brewster angle by using approximate expressions. These formulas are valid when the optical constants n and k of the surface are such that n2 + k2 ≫ 1 and are more accurate than those used so far in the literature. The excellent agreement between approximate and exact values verifies the high accuracy of the given equations.

Quantitative analysis of immunological reactions on silicon surfaces by multiple-angle Brewster angle reflectometry

An optical biosensing instrument has been developed for the quantitative analysis of immunological reactions on biochemically sensitized surfaces. It is based on the measurement of reflectance changes of polarized laser light incident on high refractive index substrates at angles close to the pseudo-Brewster angle. Multiple-angle Brewster angle reflectometry (MABAR) has been used to investigate the binding of anti-human serum albumin (a-HSA) to human serum albumin (HSA) coated silicon surfaces. The concentration dependence and reaction kinetics of antibody-antigen complex formation have been studied using red and green He-Ne laser light. A significant increase in sensitivity has been observed with green light.

Analytic solution of the pseudo-Brewster angle

The analytic solution of the pseudo-Brewster angle, for the case of light incident from a transparent medium onto an absorbing substrate, has been obtained.

A reflectance method for quantification of immunological reactions on surfaces

A simple optical method for determination of thicknesses of organic layers on solid substrates is described. The method is based on the use of a substrate with a high refractive index, e.g., silicon, and reflection of light at an angle of incidence close to the pseudo-Brewster angle. Under these conditions, a reflectance minimum is obtained for light polarized in the plane of incidence. The presence of an organic layer on the surface will increase the reflectance, which is used to determine the thickness of the layer. The physical basis of the method is described briefly, and the application to immunology is demonstrated.

Reflectometry in kinetic studies of immunological and enzymatic reactions on solid surfaces

An optical method is described, by means of which immunological and enzymatic reactions can be followed at a primary level on a solid surface, without labelling procedures. When plane-polarized light is reflected at a solid surface, there is a minimum in reflectance at a certain angle of incidence, the pseudo-Brewster angle. For example, a layer of protein adsorbed on a silicon surface increases the reflectance with increasing amount of adsorbed material. High sensitivity is obtained because of the large difference in refractive index between silicon and organic material; about 0.1 μg cm −2 adsorbed protein can be detected. In a model system of human IgG and anti-human IgG, the primary adsorption of IgG on a hydrophobic surface is first measured, and on this IgG-coated surface the binding kinetics of anti-IgG could be measured. The kinetics of proteolytic degradation of IgG-coated surfaces by trypsin was also investigated.

Pseudo-Brewster and second-Brewster angles of an absorbing substrate coated by a transparent thin film

The pseudo-Brewster angle of minimum reflectance for the p polarization, the corresponding angle for the s polarization, and the second-Brewster angle of minimum ratio of the p and s reflectances are all determined as functions of the thickness of a transparent film coating an absorbing substrate by numerical solution of the exact equations that govern such angles of the form Re(Z′/Z) = 0, where Z = Rp, Rs, or ρ represent the complex amplitude-reflection coefficients for the p and s polarizations and their ratio (ρ = Rp/Rs), respectively, and Z′ is the angle-of-incidence derivative of Z. Results that show these angles and their associated reflectance and reflectance-ratio minima are presented for the SiO2–Si film–substrate system at wavelength λ = 0.6328 μm and film thickness of up to four periods (≃1.2 μm). Applications of these results are proposed in film-thickness measurement and control.

Maximum minimum reflectance of parallel-polarized light at interfaces between transparent and absorbing media

The pseudo-Brewster angle ϕpB, of minimum reflectance ℛpm for the parallel (p) polarization, of an interface between a transparent and an absorbing medium is determined by Im{(∊ − u)[1 − (1 + ∊−1)u]2} = 0, where ∊ is the complex ratio of dielectric constants of the media and u = sin2ϕpB. It is shown that, for a given value of the normal-incidence amplitude reflectance |r|, there is an associated normal-incidence phase shift, δ = δmm, that leads to maximum minimum parallel reflectance, ℛpmm. We determine δmm, ℛpmm, ϕpBmm as functions of |r|. We find that, as |r| increases from 0 to 1, δmm decreases from 90° to 0, ℛpmm/|r|2 increases from 0 to 1, and the associated ϕpBmm decreases from 45° to 0, all monotonically.

Angle Of Incidence Reflectance Spectroscopy (Al Rs)â€â€? A Tool For Studying Optical Properties Of Solids And Liquids

Angle of Incidence Reflectance Spectroscopy (AIRS) of specular surfaces of condensed materials (solids and liquids) provides detailed information on the materials' intrinsic properties. By measuring the reflectance at one or more large angles of incidence for radiation, polarized either parallel (Rp) or perpendicular (Rs) to the plane of incidence, one gets the information necessary to solve the Fresnel relationships for the components of the complex dielectric function at the incident radiation wavelength. Experimental difficulties with absolute reflectance measurements make it desirable to measure the ratios RP /RP at a minimum of two angles of incidence. An algorithm for AIRS has been devised for use with optically isotropic materials, which requires Rp/Rs values for two angles of incidence (preferably which include the pseudo-Brewster angle OB). When a thin (d1 /A <0.05) transparent layer coats the substrate, one can solve directly for two of the four parameters (substrate indices n, k; layer index n1; and layer thickness di) given the other two. An example is given for an InGaAsP specimen where k is known, or can be reasonably estimated, as well as the value of n1. Alternatively, when n1 and d1 are known, the substrate indices can be established. The experimental apparatus is essentially the same as that for pseudo-Brewster angle mea-surements. However, because AIRS does not require the direct determination of 'PB and the ratio at that angle, the gathering of spectral data is greatly simplified.

The pseudo-Brewster angle

An exact expression for the pseudo-Brewster angle, and also some relatively easy-to-use formulas giving an approximate value for this angle and for the minimum value of the vertical polarization reflection coefficient are presented.

Optimizing precision of rotating-analyzer ellipsometers

General equations describing the theoretical uncertainty in the measurement of the complex reflectance ratio, ρ, are given for rotating-analyzer ellipsometers operating under shot-noise-limited, detector-noise-limited, and incident power-fluctuation-limited (ideal-detector) conditions. In the latter case, uncertainty is minimized when the light reflected from the sample surface is circularly polarized. For shot-noise- and detector-noise-limited configurations, minimum uncertainty is obtained as a compromise between the conditions that yield circularly polarized reflected light and those that maximize the transmitted flux. As |ρ| → 0, the uncertainty decreases linearly in |ρ| for ideal-detector systems, approaches a constant limiting value for shot-noise-limited systems, and becomes arbitrarily large for detector-noise-limited systems, which suggests that measurements at the pseudo-Brewster angle should be avoided in the latter case. A compensator is essential to achieve high precision on dielectric surfaces, but is generally not needed for metals. The theoretical relative precision to which the complex dielectric function can be obtained for silicon and gold over the visible and near-uv optical range is of the order of 1 × 10−5, which is comparable to that attainable by high-precision reflectance techniques.