Linear-to-circular polarization transformation upon reflection by a transparent thin film on a transparent substrate: analytical determination of principal angles and principal azimuths.

The principal angles and principal azimuths at which incident linearly polarized light becomes circularly polarized upon external reflection by a transparent thin film on a transparent substrate are determined analytically. For a given principal angle, multiple solutions (two or four) are obtained for the required film thicknesses and the associated principal azimuths. Specific results are presented for the air-SiO2-Si (ambient-film-substrate) system at the near-infrared wavelength λ=1.55 µm.

Interfaces with fractional optical constants and linear reflectance versus angle of incidence for incident unpolarized or circularly polarized light.

For unpolarized or circularly polarized light incident at a dielectric-conductor interface, the intensity reflectance Ru(ϕ) can be made an essentially linear function of the angle of incidence ϕ over a significant range of ϕ at specific values of the normal-incidence intensity reflectance R0 (≈1/3) and the associated normal-incidence reflection phase shift δ0 (≈40°). This places the complex refractive index n-jk of the interface in the domain of fractional optical constants. As demonstrated by specific examples, this is realizable in external reflection at vacuum-metal interfaces in the UV, and in internal reflection in the IR at interfaces between a transparent high-index substrate and an optically opaque thin film of the proper n and k. Fractional optical constants are also achievable for light reflection in air at planar surfaces of appropriately designed, nanostructured, metamaterial substrates.

Brewster-angle 50%-50% beam splitter for p-polarized infrared light using a high-index quarter-wave layer deposited on a low-index prism.

A quarter-wave layer (QWL) of high refractive index, which is deposited on a transparent prism of low refractive index, can be designed to split an incident p-polarized light beam at the Brewster angle (BA) of the air-substrate interface into p-polarized reflected and transmitted beams of equal intensity (50% each) that travel in orthogonal directions. For reflection of p-polarized light at the BA, the supported QWL functions as a free-standing (unsupported) pellicle. An exemplary design is presented that uses SixGe1-x QWL deposited on an IRTRAN1 prism for applications (such as Michelson and Mach-Zehnder interferometry) with a variable compositional fraction x in the 2-6 µm mid-IR spectral range.

Infrared Brewster-angle polarizing beam splitter using a high-index prism with a small wedge angle.

Specular reflection at the Brewster angle by a planar surface of a high-refractive-index transparent substrate is an effective means of generating linearly polarized light in the infrared. However, in such a reflection linear polarizer, the refracted light beam is usually dumped. In this paper we describe a polarizing beam splitter (PBS) that uses a high-index prism with a small wedge angle to generate two orthogonally p and s linearly polarized beams that travel in orthogonal directions. A single-layer antireflection coating (ARC) at the exit face of the prism can be embedded with aligned Ag nano needles to suppress the transmission of the small, residual, s-polarized refracted component. An example of such PBS that uses a PbTe prism with ZnSe ARC in the mid-infrared is presented.

Bare and thin-film-coated substrates with null reflection for p- and s-polarized light at the same angle of incidence: reflectance and ellipsometric parameters as functions of substrate refractive index and film thickness.

Intensity reflectances and ellipsometric parameters of a partially clad transparent substrate that suppresses the reflection of incident p- and s-polarized light at the same angle of incidence from uncoated and single-layer-coated areas are determined as functions of normalized film thickness ς and substrate refractive index n2. The common polarizing angle is the Brewster angle of the ambient-substrate interface, and the light beam incident from the ambient (air or vacuum) is refracted in the film at a 45° angle from the normal to the parallel-plane film boundaries. For n2≤2, the differential reflection phase shift Δ=δp-δs≈±90° for all values of ς so that the Brewster angle is also approximately the principal angle of the film-substrate system independent of film thickness. Accurate techniques for monitoring the deposition of such films are also proposed.

Stokes-vector and Mueller-matrix polarimetry [Invited].

This paper reviews the current status of instruments for measuring the full 4×1 Stokes vector S, which describes the state of polarization (SOP) of totally or partially polarized light, and the 4×4 Mueller matrix M, which determines how the SOP is transformed as light interacts with a material sample or an optical element or system. The principle of operation of each instrument is briefly explained by using the Stokes-Mueller calculus. The development of fast, automated, imaging, and spectroscopic instruments over the last 50 years has greatly expanded the range of applications of optical polarimetry and ellipsometry in almost every branch of science and technology. Current challenges and future directions of this important branch of optics are also discussed.

Angularly symmetric splitting of a light beam upon reflection and refraction at an air-dielectric plane boundary: reply.

The simplified explicit expressions derived by Andersen [J. Opt. Soc. Am. A33, 984 (2016)JOAOD60740-323210.1364/JOSAA.32.000984], that relate to angularly symmetric beam splitting by reflection and refraction at an air-dielectric interface recently described by Azzam [J. Opt. Soc. Am. A32, 2436 (2015)JOAOD60740-323210.1364/JOSAA.32.002436], are welcome. A few additional remarks are also included in my reply to Andersen's comment.

Difference between the Brewster angle and angle of minimum reflectance for incident unpolarized or circularly polarized light at interfaces between transparent media.

For reflection at interfaces between transparent optically isotropic media, the difference between the Brewster angle ϕB of zero reflectance for incident p-polarized light and the angle ϕu min of minimum reflectance for incident unpolarized or circularly polarized light is considered as function of the relative refractive n in external and internal reflection. We determine the following. (i) ϕu min < ϕB for all values of n. (ii) In external reflection (n > 1), the maximum difference (ϕB - ϕu min)max = 75° at n = 2 + √3. (iii) In internal reflection and 0 < n ≤ 2 - √3, (ϕB - ϕu min)max = 15° at n = 2 - √3; for 2 - √3 < n < 1, ϕu min = 0, and (ϕB - ϕu min)max = 45° as n → 1. (iv) For 2 - √3 ≤ n ≤ 2 + √3, the intensity reflectance R0 at normal incidence is in the range 0 ≤ R0 ≤ 1/3, ϕu min = 0, and ϕB - ϕu min = ϕB. (v) For internal reflection and 0 < n < 2 - √3, ϕu min exhibits an unexpected maximum (= 12.30°) at n = 0.24265. Finally, (vi) for 1/3 ≤ R0 < 1, Ru min at ϕu min is limited to the range 1/3 ≤ Ru min < 1/2.

Necessary and sufficient conditions for the appearance of a reflectance minimum at oblique incidence when unpolarized or circularly polarized light is reflected at a dielectric-conductor interface.

The appearance of a reflectance minimum at oblique incidence when unpolarized or circularly polarized light is reflected at a dielectric-conductor interface requires that the normal-incidence intensity reflectance R(0) of the interface be >1/3 [J. Opt. Soc. A9, 957 (1992)10.1364/JOSAA.9.000957; Appl. Opt.53, 7885 (2014)APOPAI0003-693510.1364/AO.53.007885]. However, R(0) 1/3 is only a necessary but insufficient condition for the interface reflectance to exhibit a minimum at non-normal incidence. A second condition, the subject of this study, restricts the normal-incidence reflection phase shift δ(0) for s-polarized light to one of two non-overlapping bands: (a) 0≤δ(0)<δ(0 max) and (b) δ(0 min)<δ(0)≤180°. These two bands are associated with internal and external reflection, respectively. The limiting phase shifts δ(0 max) and δ(0 min) at the band edges are determined analytically as functions of R(0) .

Angularly symmetric splitting of a light beam upon reflection and refraction at an air-dielectric plane boundary.

Conditions for achieving equal and opposite angular deflections of a light beam by reflection and refraction at an air-dielectric boundary are determined. Such angularly symmetric beam splitting (ASBS) is possible only if the angle of incidence is >60° by exactly one third of the angle of refraction. This simple law, plus Snell's law, leads to several analytical results that clarify all aspects of this phenomenon. In particular, it is shown that the intensities of the two symmetrically deflected beams can be equalized by proper choice of the prism refractive index and the azimuth of incident linearly polarized light. ASBS enables a geometrically attractive layout of optical systems that employ multiple prism beam splitters.

Beam splitters for p-polarized light using a high-index quarter-wave layer embedded in a low-index cube prism.

A high-index quarter-wave layer (QWL) embedded in a low-index cube prism is designed to achieve 50%-50% beam splitting for incident p-polarized light at a 45° angle of incidence. This is accomplished when the ratio of the refractive index of the QWL to that of the prism is n=3.336666. Such a refractive index ratio is realized, e.g., with a Ge QWL embedded in a LiF cube at 8.357 µm wavelength. Spectral, angular, and film-thickness sensitivities of this mid-IR beam splitter (BS) are presented. Free-standing QWL pellicles of GaP and GaAs can also function as 50%-50% BSs for incident p-polarized light at 45° at visible and IR wavelengths of 0.610 µm and 2.929 µm, respectively. An application in interferometry is briefly discussed.

Maximum reflectance difference for incident p- and s-polarized light at air-dielectric interfaces.

New features related to the reflection of p- and s-polarized light at optically isotropic air-dielectric interfaces are determined. The intensity reflectance difference, RD = Rs - Rp, is maximum at an angle of incidence (AOI) ϕ = ϕMD in the range between the Brewster angle and grazing incidence. Both ϕMD and (Rs - Rp)max are calculated as functions of substrate refractive index n. Explicit expressions are obtained for the two angles of incidence, one below and one above the Brewster angle, at which the condition Rs = 2Rp is satisfied. Measurement of such angles enables the determination of n of transparent materials using a simple null-seeking spectroscopic ellipsometer with a ϕ-2ϕ goniometer. Finally, the AOI at which Rp'' = 0 (point of inflection in the Rp-versus-ϕ curve) is identified, and the equation that determines such an angle is also derived analytically.

Angle of incidence of minimum reflectance of a dielectric-conductor interface for incident unpolarized or circularly polarized light.

The angle of incidence ϕ=ϕu min of minimum reflectance for incident unpolarized or circularly polarized light at a dielectric-conductor interface is determined for any complex relative refractive index N=(n,k), and contours of constant ϕu min in the nk plane are presented. The minimum reflectance Ru min at ϕu min is also plotted as a function of the polar angle 0≤θ=arg(N)≤90° along each constant ϕu min contour. Also presented are families of Ru-versus-ϕ curves for values of complex N at θ=30°, ϕu min=45° to 85° in steps of 10°, and values of complex N at ϕu min=75°, θ=0° to 90° in steps of 10°. Finally, a nonpolarimetric method for the determination of n and k of optical materials, which is based on measurements of ϕu min and the normal-incidence reflectance R0, is proposed.

Complex reflection coefficients of p- and s-polarized light at the pseudo-Brewster angle of a dielectric-conductor interface.

The complex Fresnel reflection coefficients r(p) and r(s) of p- and s-polarized light and their ratio ρ=r(p)/r(s) at the pseudo-Brewster angle (PBA) φ(pB) of a dielectric-conductor interface are evaluated for all possible values of the complex relative dielectric function ε=|ε|exp(-jθ)=ε(r)-jε(i), ε(i)>0 that share the same φ(pB). Complex-plane trajectories of r(p), r(s), and ρ at the PBA are presented at discrete values of φ(pB) from 5° to 85° in equal steps of 5° as θ is increased from 0° to 180°. It is shown that for φ(pB)>70° (high-reflectance metals in the IR) r(p) at the PBA is essentially pure negative imaginary and the reflection phase shift δ(p)=arg(r(p))≈-90°. In the domain of fractional optical constants (vacuum UV or light incidence from a high-refractive-index immersion medium) 0<φ(pB)<45° and r(p) is pure real negative (δ(p)=π) when θ=tan(-1)(√(cos(2φ(pB)))), and the corresponding locus of ε in the complex plane is obtained. In the limit of ε(i)=0, ε(r)<0 (interface between a dielectric and plasmonic medium) the total reflection phase shifts δ(p), δ(s), Δ=δ(p)-δ(s)=arg(ρ) are also determined as functions of φ(pB).

Three-dimensional polarization states of monochromatic light fields.

The 3×1 generalized Jones vectors (GJVs) [E(x) E(y) E(z)](t) (t indicates the transpose) that describe the linear, circular, and elliptical polarization states of an arbitrary three-dimensional (3-D) monochromatic light field are determined in terms of the geometrical parameters of the 3-D vibration of the time-harmonic electric field. In three dimensions, there are as many distinct linear polarization states as there are points on the surface of a hemisphere, and the number of distinct 3-D circular polarization states equals that of all two-dimensional (2-D) polarization states on the Poincaré sphere, of which only two are circular states. The subset of 3-D polarization states that results from the superposition of three mutually orthogonal x, y, and z field components of equal amplitude is considered as a function of their relative phases. Interesting contours of equal ellipticity and equal inclination of the normal to the polarization ellipse with respect to the x axis are obtained in 2-D phase space. Finally, the 3×3 generalized Jones calculus, in which elastic scattering (e.g., by a nano-object in the near field) is characterized by the 3-D linear transformation E(s)=T E(i), is briefly introduced. In such a matrix transformation, E(i) and E(s) are the 3×1 GJVs of the incident and scattered waves and T is the 3×3 generalized Jones matrix of the scatterer at a given frequency and for given directions of incidence and scattering.

Circular and near-circular polarization states of evanescent monochromatic light fields in total internal reflection.

Conditions for the production of near-circular polarization states of the evanescent field present in the rarer medium in total internal reflection of incident monochromatic p-polarized light at a dielectric-dielectric planar interface are determined. Such conditions are satisfied if high-index (>3.2) transparent prism materials (e.g., GaP and Ge) are used at angles of incidence well above the critical angle but sufficiently below grazing incidence. Furthermore, elliptical polarization of incident light with nonzero p and s components can be tailored to cause circular polarization of the resultant tangential electric field in the plane of the interface or circular polarization of the transverse electric field in a plane normal to the direction of propagation of the evanescent wave. Such polarization control of the evanescent field is significant, e.g., in the fluorescent excitation of molecules adsorbed at solid-liquid and solid-gas interfaces by total internal reflection.

Simplified design of thin-film polarizing beam splitter using embedded symmetric trilayer stack.

An analytically tractable design procedure is presented for a polarizing beam splitter (PBS) that uses frustrated total internal reflection and optical tunneling by a symmetric LHL trilayer thin-film stack embedded in a high-index prism. Considerable simplification arises when the refractive index of the high-index center layer H matches the refractive index of the prism and its thickness is quarter-wave. This leads to a cube design in which zero reflection for the p polarization is achieved at a 45° angle of incidence independent of the thicknesses of the identical symmetric low-index tunnel layers L and L. Arbitrarily high reflectance for the s polarization is obtained at subwavelength thicknesses of the tunnel layers. This is illustrated by an IR Si-cube PBS that uses an embedded ZnS-Si-ZnS trilayer stack.

Principal angles and principal azimuths of frustrated total internal reflection and optical tunneling by an embedded low-index thin film.

The condition for obtaining a differential (or ellipsometric) quarter-wave retardation when p- and s-polarized light of wavelength λ experience frustrated total internal reflection (FTIR) and optical tunneling at angles of incidence ϕ ≥ the critical angle by a transparent thin film (medium 1) of low refractive index n1 and uniform thickness d, which is embedded in a transparent bulk medium 0 of high refractive index n0 takes the simple form: -tanh2 x = tan δp tan δs, in which x = 2πn1(d/λ)(N2sin2ϕ - 1)(1/2), N = n0/n1, and δp, δs are 01 interface Fresnel reflection phase shifts for the p and s polarizations. From this condition, the ranges of the principal angle and normalized film thickness d/λ are obtained explicitly. At a given principal angle, the associated principal azimuths ψr, ψt in reflection and transmission are determined by tan2ψr = -sin 2δs/sin 2δp and tan2ψt = -tan δp/tan δs, respectively. At a unique principal angle ϕe given by sin2ϕe = 2/(N2 + 1), ψr = ψt = 45° and linear-to-circular polarization conversion is achieved upon FTIR and optical tunneling simultaneously. The intensity transmittances of p- and s-polarized light at any principal angle are given by τp = tan δp/tan (δp - δs) and τs = -tan δs/tan (δp - δs), respectively. The efficiency of linear-to-circular polarization conversion in optical tunneling is maximum at ϕe.

Return-path, multiple-principal-angle, internal-reflection ellipsometer for measuring IR optical properties of aqueous solutions.

A retroreflection (return-path) spectroscopic ellipsometer without a wave plate is described that uses an IR-transparent high-refractive-index hemicylindrical semiconductor substrate to measure the optical properties of aqueous solutions from multiple principal angles and multiple principal azimuths of attenuated internal reflection (AIR) at the semiconductor-solution interface. The pseudo-Brewster angle of minimum reflectance for the p polarization is also readily measured using the same instrument. This wealth of data can also be used to characterize thin films at the solid-liquid interface. Simulated results of AIR at the Si-water interface over the 1.2-11?mum IR spectral range are presented in support of this concept. The optical properties of water and aqueous solutions are important for modeling radiative transfer in the atmosphere and oceans and for biomedical and tissue optics.