Analogy Between Linear Optical Systems and Linear Two-Port Electrical Networks

Attention is called to the analogy between linear optical systems and linear two-port electrical networks. For both, the transformation of a pair of oscillating quantities between input and output is of interest. The mapping of polarization by an optical system and of impedance (admittance) by a two-port network is described by a bilinear transformation. Therefore for each transfer property of a system of one type, there is a similar property for the system of the other type. Two-port electrical networks are synthesized whose impedance-(or admittance-) mapping properties are the same as the polarization-mapping properties of a given optical system. The opposite problem of finding the optical analogs of two-port networks is also considered. Besides unifying the methods of handling these two different kinds of systems, the analogy appears fruitful if used reciprocally to simulate electrical networks by optical systems, and vice versa. Linear mechanoacoustic systems have optical analogs besides their well-known electrical analogs

Analogy Between Linear Optical Systems and Linear Two-Port Electrical Networks

Attention is called to the analogy between linear optical systems and linear two-port electrical networks. For both, the transformation of a pair of oscillating quantities between input and output is of interest. The mapping of polarization by an optical system and of impedance (admittance) by a two-port network is described by a bilinear transformation. Therefore for each transfer property of a system of one type, there is a similar property for the system of the other type. Two-port electrical networks are synthesized whose impedance-(or admittance-) mapping properties are the same as the polarization-mapping properties of a given optical system. The opposite problem of finding the optical analogs of two-port networks is also considered. Besides unifying the methods of handling these two different kinds of systems, the analogy appears fruitful if used reciprocally to simulate electrical networks by optical systems, and vice versa. Linear mechanoacoustic systems have optical analogs besides their well-known electrical analogs

Loci of Invariant-Azimuth and Invariant-Ellipticity Polarization States of an Optical System

The loci of polarization states for which either the ellipticity alone or the azimuth alone remains invariant upon passing through an optical system are introduced. The cartesian equations of these two loci are derived in the complex plane in which the polarization states are represented. The equations are quartic and are conveniently expressed in terms of the elements of the Jones. matrix of the optical system. As an exple the loci are determined for a system composed of a π/4 rotator followed by a quarter-wave retarder

Design of film-substrate single-reflection linear partial polarizers

The results of a preceding paper [J. Opt. Soc. Am. 65, 1464,(1975)] are viewed from a different angleas providing the basis for the design of film-substrate single-reflection linear partial polarizers (LPP),which also operate as reflection optical rotators. The important characteristics of a comprehensive set of discrete designs of SiO2-Si LPP’s at λ = 6328 Å are shown graphically

Polarizer-surface-analyzer null ellipsometry for film-substrate systems

Single-pass polarizer-surface-analyzer null ellipsometry (PSA-NE) can be used to characterize film-substrate systems, provided that the film thickness lies within one of a set of permissible-thickness bands (PTB). For a transparent film on a transparent or absorbing substrate, the PTB structure consists of a small number of finite-bandwidth bands followed by a continuum band that extends from a film thickness of about half the wavelength of light to infinity. We show that this band structure is a direct consequence of the periodicity of the ellipsometric function ρ (the ratio Rp/Rs, of the complex amplitude-reflection coefficients for the p and s polarizations) with film thickness. The PTB for the SiO2-Si film-substrate system at He-Ne laser and mercury spectral lines are calculated. The angles of incidence for PSA-NE ona film-substrate system with known film thickness are easily predicted with the help of a graphical construction in the angle of incidence-vs-thickness φdplane. PSA-NE is generally applicable to the determination of both film thickness and optical properties of a film-substrate system. The procedure for its application to the special, but important, case of film-thickness measurement alone, when the optical properties are known, is given and is checked experimentally by the determination of the oxide-film thickness on Si wafers. In an automated form, PSA-NE can be a serious competitor for interferometric reflectance methods

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

Design of film—substrate single-reflection retarders

The design steps for film—substrate single-reflection retarders are briefly stated and applied to the SiO2—Si film—substrate system at wavelength 6328 Å. The criterion of minimum-maximum error of the ellipsometric angle ψ is used to choose angle-of-incidence-tunable designs. Use is made of the (Φ-d) plane (angle of incidence versus thickness) to determine whether a given film—substrate system with known optical properties and film thickness can operate as a reflection retarder and to determine the associated angles of incidence and retardation angles. This leads to the concept of permissible-thickness bands and forbidden gaps for operation of a film—substrate system as a reflection retarder. Experimental measurements on one of the proposed designs proved the validity of the method

Ellipsometric function of a film-substrate system: Applications to the design of reflection-type optical devices and to ellipsometry

The ratio ρ = Rp/Rs of the complex amplitude-reflection coefficients Rp and Rs for light polarized parallel (p) and perpendicular (s) to the plane of incidence, reflected from an optically isotropic film-substrate system, is investigated as a function of the angle of incidence ϕ and the film thickness d. Both constant-angle-of-incidence contours (CAIC) and constant-thickness contours (CTC) of the ellipsometric function ρ(ϕ,d) in the complex ρ plane are examined. For transparent films, ρ(ϕ,d) is a periodic function of d with period Dϕ that is a function of ϕ. For a given angle of incidence ϕ and film thickness d (0 ≤ ϕ ≤ 90, 0 ≤ d ≤ Dϕ), the equispaced linear array of points (ϕ,d + mDϕ) (m = 0, 1, 2,…) in the real (ϕ,d) plane is mapped by the periodic function ρ(ϕ,d) into one distinct point in the complex ρ plane. Conversely, for a given film-substrate system, any value of the ellipsometric function ρ can be realized at one particular angle of incidence ϕ and an associated infinite series of film thicknesses d, d + Dϕ, d + 2Dϕ,.... This analysis leads to (1) a unified scheme for the design of all reflection-type optical devices, such as polarizers and retarders, (2) a novel null ellipsometer without a compensator, for the measurement of films whose thicknesses are within certain permissible ranges, and (3) an inversion procedure for the nonlinear equations of reflection ellipsometry that separates the determination of the optical constants (refractive indices and extinction coefficients) of the film and substrate from the film thickness. Extension of the work to absorbing films is discussed