Modulated generalized ellipsometry

We extend ellipsometry to the direct measurement of small perturbations of the Jones matrix of any linear nondepolarizing optical sample (system) subjected to a modulating stimulus such as temperature, stress, or electric or magnetic field. The methodology of this technique, to be called Modulated Generalized Ellipsometry (MGE), is presented. First an ellipsometer with arbitrary polarizing and analyzing optics is assumed, and subsequently the discussion is specialized to a conventional ellipsometer having either the polarizer-sample-analyzer (PSA) or the polarizer-compensator-sample-analyzer (PCSA) arrangement. MGE provides the tool for the systematic study of thermo-optical, piezo-optical, electro-optical, magneto-optical, and other allied effects for both isotropic and anisotropic materials that may be examined in either transmission or reflection. MGE is also applicable to (1) modulation spectroscopy of anisotropic media, (2) the study of electrochemical reactions on optically anisotropic electrodes, and (3) the extension of AIDER (angle-of-incidence-derivative ellipsometry and reflectometry) to the determination of the optical properties of anisotropic film-substrate systems

Reflection of an electromagnetic plane wave with 0 or π phase shift at the surface of an absorbing medium

An electromagnetic plane wave incident obliquely from a transparent medium onto the surface of an absorbing medium can be reflected with 0 or π phase shift if (i) the wave is p (TM) polarized, and (ii) the complex relative dielectric function ε is such that 0 ≤|ε| 2/2Re(ε) ≤ 1. Furthermore, the locus of ε such that the reflection coefficient for the p polarization is real at the same angle of incidence, is a circle, and that of ε½ (the complex relative refractive index) is Bernoulli’s lemniscate

Thin-film beam splitter that reflects light as a half-wave retarder and transmits it without change of polarization: application to a Michelson interferometer

The refractive index n1 of a transparent layer of quarter-wave optical thickness coating a transparent substrate of refractive index n2 can be chosen to produce half-wave retardation (HWR) in reflection and no change of polarization in refraction at any angle of incidence ø. The function n1(ø, n2), and the associated polarization-independent reflectance of the film-substrate system R(ø, n2) are determined. Such a coated surface can be used as a beam splitter with excellent characteristics (e.g., split fractions that do not depend on source polarization, a split beam whose polarization is identical to that of the incident beam and operation over a wide range of incidence angles). A concrete example of a coated Ge-slab beam splitter for 10.6-µm radiation at ø = 45° is given. The beam-splitter face of the slab is coated with the HWR layer, and the exit face is coated with a double layer that produces total refraction without change of polarization. Such a beam splitter is tolerant to film-thickness errors and is reasonably achromatic over a small (e.g., 10–11-µm) wavelength range. When used in a Michelson interferometer this beam splitter renders its operation totally independent of source polarization

Single-layer antireflection coatings on absorbing substrates for the parallel and perpendicular polarizations at oblique incidence

Explicit equations are derived that determine the refractive index of a single layer that suppresses the reflection of p- or s-polarized light from the planar interface between a transparent and an absorbing medium at any given angle of incidence. The required layer thickness and the system reflectance for the orthogonal unextinguished polarization also follow explicitly. This generalizes earlier work that was limited to normal incidence or to oblique incidence at dielectric—dielectric interfaces. Specific examples are given of p- and s-antireflection layers on Si and Al substrates at λ = 6328 Å at various angles of incidence

Propagation of partially polarized light through anisotropic media with or without depolarization: A differential 4 × 4 matrix calculus

We extend the scope of the Mueller calculus to parallel that established by Jones for his calculus. We find that the Stokes vector S of a light beam that propagates through a linear depolarizing anisotropic medium obeys the first-order linear differential equation dS/dz = mS, where z is the distance traveled along the direction of propagation and m is a 4 × 4 real matrix that summarizes the optical properties of the medium which influence the Stokes vector. We determine the differential matrix m for eight basic types of optical behavior, find its form for the most general anisotropic nondepolarizing medium, and determine its relationship to the complex 2 × 2 differential Jones matrix. We solve the Stokes-vector differential equation for light propagation in homogeneous nondepolarizating media with arbitrary absorptive and refractive anisotropy. In the process, we solve the differential-matrix and Mueller-matrix eigenvalue equations. To illustrate the case of inhomogeneous anisotropic media, we consider the propagation of partially polarized light along the helical axis of a cholesteric or twisted-nematic liquid crystal. As an example of depolarizing media, we consider light propagation through a medium that tends to equalize the preference of the state of polarization to the right and left circular states

Transmission ellipsometry on transparent unbacked or embedded thin films with application to soap films in air

The ratio ρt = Tp/Ts of the complex amplitude transmission coefficients for the p and s polarizations of a transparent unbacked or embedded thin film is examined as a function of the film thickness-to-wavelength ratio d/λ and the angle of incidence Φ for a given film refractive index N. The maximum value of the differential transmission phase shift (or retardance), Δt = argρt, is determined, for given N and Φ, by a simple geometrical construction that involves the iso-Φ circle locus of ρt in the complex plane. The upper bound on this maximum equals arctan{[N - (1/N)]/2} and is attained in the limit of grazing incidence. An analytical noniterative method is developed for determining N and d of the film from ρt measured by transmission ellipsometry (TELL) at Φ = 45°. An explicit expression for d Δt of an ultrathin film, d/λ « 1, is derived in product form that shows the dependence of Δt on N, Φ, and d/λ separately. The angular dependence is given by an obliquity factor, f0(Φ) = 2½ sinΦ tanΦ, which is verified experimentally by TELL measurements on a stable planar soap film in air at λ = 633 nm. The singularity of f0 at Φ = 90° is resolved; Δt is shown to have a aximum just short of grazing incidence and drops to 0 at Φ = 90°. Because N and d/λ are inseparable for an ultrathin film, N is determined by a Brewster angle measurement and d/λis subsequently obtained from Δt Finally, the ellipsometric function in reflection ρr is related to that in transmission Pt

Relationship between the p and s Fresnel reflection coefficients of an interface independent of angle of incidence

The Fresnel reflection coefficients rp and rs of p- and s-polarized light at the planar interface between two linear isotropic media are found to be interrelated by (rs - rp)/(1 - rsrp) = cos 2β, independent of the angle of incidence ø, where tan2β = ∊ and ∊ is the (generally complex) ratio of dielectric constants of the media of refraction and incidence. This complements another relation (found earlier), (r2s - rp)/(rs - rsrp) = cos 2φ, which is valid at a given ø independent of ∊ (i.e., for all possible interfaces). Taken together, these two equations specify rp and rs completely and can be used to replace the original Fresnel equations

Simple and direct determination of complex refractive index and thickness of unsupported or embedded thin films by combined reflection and transmission ellipsometry at 45° angle of incidence

Measurements of the polarization states (represented by complex numbers Xr and Xt, respectively) of light reflected and transmitted by an unsupported or embedded thin film, for totally polarized light (with nonzero p and s components) incident at 45°, permit simple, direct, and explicit determination of the film\u27s complex refractive index N1 independently of film thickness or input polarization. If α = Xr/Xt, we find that α = rs + rs-1, where rs is Fresnel’s complex reflection coefficient of the ambient-film interface for the s polarization at 45° incidence. From α, rs is determined, and from rs we get N1 = N0(1 + rs2)1/2/(1 + rs), where N0 is the refractive index of the transparent medium surrounding the film. Knowledge of the incident polarization Xi allows the film thickness to be determined, also explicitly, by using either of the ratios Xi/Xr or Xi/Xt