ScholarWorks@UNO

Casting No Shadow: Incarceration, Architecture, and Love in Canto 90

John Gery — Tue, 15 May 2012 09:40:03 PDT

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

R. M.A. Azzam — Fri, 11 May 2012 13:39:11 PDT

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.

Antireflecting and polarizing transparent bilayer coatings on absorbing substrates at oblique incidence

R. M.A. Azzam et al. — Fri, 11 May 2012 13:39:09 PDT

The condition of zero reflection of p- and s-polarized light by a transparent bilayer on an absorbing substrate is derived in the form |g_ν(ø,N_i)| ≤ 1, where g_ν is a function of the angle of incidence ø, the refractive indices N_i(i = 0,1,2,3) of the system, and the polarization state ν (= p or s). As an application, the air-Si₃N₄-SiO₂-Si system is considered at two laser wavelengths λ = 6328 and 3250 Å. The thicknesses of the two films of the bilayer and the unextinguished reflectance are determined as functions of ø, and the results appear graphically and in tables. Extinction of the s polarization is accompanied by low overall residual reflectance (e.g., for incident unpolarized light, it is 1.6% for λ = 6328 Å at ø = 45°). On the other hand, suppression of the p polarization at a high incidence angle is accompanied by high s reflectance (e.g. = 96%for λ = 3250 Å at ø = 83°). This demonstrates that efficient bilayer reflection polarizers are possible.

Division-of-wave-front polarizing beam splitter and half-shade device using dielectric thin film on dielectric substrate

R. M.A. Azzam — Fri, 11 May 2012 13:39:08 PDT

Three-reflection halfwave and quarterwave retarders using dielectric-coated metallic mirrors

T. F. Thonn et al. — Fri, 11 May 2012 13:39:07 PDT

A design procedure is described to determine the thicknesses of single-layer coatings of a given dielectric on a given metallic substrate so that a specified net phase retardance (and/or a net relative amplitude attenuation) between the p and s polarizations is achieved after three reflections from a symmetrical arrangement of three mirrors that maintain collinearity of the input and output beams. Examples are presented of halfwave and quarterwave retarders (HWR and QWR) that use a ZnS-Ag film-substrate system at the CO₂-laser wavelength λ = 10.6 µm. The equal net reflectances for the p and s polarizations are computed and found to be high (above 90%) for most designs. Sensitivity of the designs (deviation of the magnitude and phase of the ratio of net complex p and s reflection coefficients from design specifications) to small film thickness and angle-of-incidence errors is examined, and useful operation over a small wavelength range (10–11 µm) is demonstrated.

Multiple determination of the optical constants of thin-film coating materials

D. P. Arndt et al. — Fri, 11 May 2012 13:39:06 PDT

The seven participating laboratories received films of two different thicknesses of Sc₂O₃ and Rh. All samples of each material were prepared in a single deposition run. Brief descriptions are given of the various methods used for determination of the optical constants of these coating materials. The measurement data are presented, and the results are compared. The mean of the variances of the Sc₂O₃refractive-index determinations in the 0.40–0.75-nm spectral region was 0.03. The corresponding variances for the refractive index and absorption coefficient of Rh were 0.35 and 0.26, respectively.

Complex reflection coefficients for the parallel and perpendicular polarizations of a film-substrate system

R. M.A. Azzam et al. — Fri, 11 May 2012 13:39:05 PDT

The complex reflection coefficients R_v(ø,ζ) of a film-substrate system for the parallel (v = p) and perpendicular (v = s) polarizations are examined in detail as functions of the angle of incidence ø(0 ≤ ø ≤ 90°) and the reduced normalized film thickness ζ(0 ≤ ζ < 1). For definiteness, the reflection of light of wavelength λ = 0.6328 µm by the air–SiO₂–Si system is assumed. Families of circles that represent the constant-angle-of-incidence contours, their envelopes, and the associated constant-thickness contours ofR_p and R_s are all presented in the complex plane. Furthermore, the amplitude-reflectance and phase-shift functions, |R_v|(ø,ζ) and argR_v(ø,ζ) are plotted vs ζ with ø constant and vs ø with ζ constant. It is shown that R_p or R_s can assume the same complex value at two different angles of incidence (i.e., the film-substrate system can have identical reflection characteristics for a given polarization at two angles) for certain ranges of film thickness. The distinct case of internal reflection is represented by a separate example.

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

R. M.A. Azzam et al. — Fri, 11 May 2012 13:39:04 PDT

The pseudo-Brewster angle of minimum reflectance for the p polarization, the corresponding angle for thes 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 = R_p, R_s, or ρ represent the complex amplitude-reflection coefficients for the p and s polarizations and their ratio (ρ =R_p/R_s), 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-sibstrate 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.

Constant-psi constant-delta contour maps: applications to ellipsometry and to reflection-type optical devices

A.-R. M. Zaghloul et al. — Fri, 11 May 2012 13:39:03 PDT

Constant-psi constant-delta contour maps in the reduced angle-of-incidence-film-thickness plane that are useful in ellipsometry and in design of reflection-type optical devices are discussed. As a specific example, a contour map is given for the SiO₂-Si film-substrate system at the 6328-Å He-Ne laser wavelength.

Scheme to polarization-correct a waxicon

R. M.A. Azzam — Fri, 11 May 2012 13:39:01 PDT

Polarization-preserving single-layer-coated beam displacers and axicons

R. M.A. Azzam et al. — Fri, 11 May 2012 13:39:00 PDT

A design procedure is described to determine the thicknesses of single-layer coatings of a given dielectric on a given metallic substrate so that any input polarization of light is preserved after two reflections at the same angle of incidence from a parallel-mirror beam displacer or an axicon. This is achieved by equalizing the net complex p and s reflection coefficients (also called the radial and azimuthal eigenvalues of an axicon) after two reflections. The net polarization-independent reflectance (insertion loss) of the device is computed and found to exceed the net minimum parallel reflectance of the uncoated device for incidence above a certain angle. Thus the dielectric films serve (1) protective, (2) polarization-corrective, and (3) reflectance-enhancement functions simultaneously. The sensitivity of the designs (deviation from the condition of polarization preservation) to small film-thickness and angle-of-incidence errors is examined. Results are presented graphically and in tables of applications of the method to beam displacers and axicons for He-Ne laser light, λ = 0.6328 µm, using Al₂O₃ (or Si₂O₃)/Al and MgF₂/Al film-substrate systems, and for CO₂ laser light, λ = 10.6 µm, using ZnS or ThF₄ films on an Ag substrate.

Explicit determination of thickness of a transparent film on a transparent substrate from angles of incidence of equal p and s reflectivities

R. M.A. Azzam — Fri, 11 May 2012 13:38:59 PDT

Relations between amplitude reflectances and phase shifts of the p and s polarizations when electromagnetic radiation strikes interfaces between transparent media

R. M.A. Azzam — Fri, 11 May 2012 13:38:56 PDT

SiO2-Si film-substrate reflection polarizers for different mercury spectral lines

A.-R. M. Zaghloul et al. — Fri, 11 May 2012 13:38:55 PDT

Three-dimensional polarization states of monochromatic light fields

R. M.A. Azzam — Fri, 11 May 2012 13:38:52 PDT

The 3×1 generalized Jones vectors (GJVs) [ExEyEz]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 Es=T Ei , is briefly introduced. In such a matrix transformation, Ei and Es 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.

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

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:51 PDT

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: −tanh2x=tanδptanδs , in whichx=2πn1(d/λ)(N2sin2ϕ−1)1/2 , N=n0/n1 , and δp , δs are 01 interface Fresnel reflection phase shifts for the pand 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=−sin2δs/sin2δp and tan2ψt=−tanδp/tanδs , respectively. At a unique principal angle ϕegiven 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 byτp=tanδp/tan(δp−δs) and τs=−tanδs/tan(δp−δs) , respectively. The efficiency of linear-to-circular polarization conversion in optical tunneling is maximum at ϕe .

Transmission of p- and s-polarized light through a prism and the condition of minimum deviation

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:50 PDT

The condition of minimum deviation (MD) by a transparent optically isotropic prism is re-derived, and expressions for the intensity transmittances Tp(θ) and Ts(θ) of an uncoated prism of refractive index n and prism angle α for incident p- ands-polarized light and their derivatives with respect to the internal angle of refraction θ are obtained. When the MD condition(θ=α/2) is satisfied, Ts is maximum and Tp is maximum or minimum. The transmission ellipsometric parametersψt,Δt of a symmetrically coated prism are also shown to be locally stationary with respect to θ at θ=α/2 . The constraint on (n,α) for maximally flat transmittance (MFT) of p-polarized light at and near the MD condition is determined. The transmittance Tp of prisms represented by points that lie below the locus (n,α) of MFT exhibits oscillation as a function of θ. No similar behavior is found for the s polarization. Magnitudes and angular positions of the maxima and minima of the oscillatory Tp -versus-θ curves are also calculated as functions of α for a ZnS prism of refractive indexn=2.35 in the visible.

Difference between the second-Brewster and pseudo-Brewster angles when polarized light is reflected at a dielectric–conductor interface

A. Alsamman et al. — Fri, 11 May 2012 13:38:49 PDT

For a given pseudo-Brewster angle ϕpB of minimum reflectance ∣∣rp∣∣ of p-polarized light at a dielectric-conductor interface, the second-Brewster angle ϕ2B of minimum reflectance ratio |ρ|=∣∣rp∣∣/|rs| of the p and s polarizations is determined for all possible values of the complex relative dielectric function ϵ that lead to the same ϕpB . The difference ϕ2B−ϕpB is considered as a function of ϕpB and θ=arg(ϵ) . For any given ϕpB , the difference ϕ2B−ϕpB=0 atθ=0(ϵr>0,ϵi=0) increases monotonically as a function of θ and reaches maximum value {ϕ2B−ϕpB}max in the limit as θ→180° (ϵr<0,ϵi=0) . This maximum difference {ϕ2B−ϕpB}max has an upper limit of 15.701° whenϕpB=28.195° .

Plurality of principal angles for a given pseudo-Brewster angle when polarized light is reflected at a dielectric-conductor interface

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:48 PDT

The pseudo-Brewster angle ϕpB of minimum reflectance for p-polarized light and the principal angle ϕ¯ at which incident linearly polarized light of the proper azimuth is reflected circularly polarized are considered as functions of the complex relative dielectric function ε of a dielectric–conductor interface over the entire complex ε plane. In particular, the spread of ϕ¯for a given ϕpB is determined, and the maximum difference (ϕ¯−ϕpB)max is obtained as a function of ϕpB . The maximum difference (ϕ¯−ϕpB)max approaches 45° and 0 in the limit as ϕpB→0 and 90°, respectively. ForϕpB<22.666° , multiple principal angles ϕ¯i , i=1,2,3 , appear for each ε in a subdomain of fractional optical constants. This leads to an elaborate pattern of multiple solution branches for the difference ϕ¯i−ϕpB , i=1,2,3 , as is illustrated by several examples.

Efficiency of linear-to-circular polarization conversion for light reflection at the principal angle by a dielectric-conductor interface

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:47 PDT

The efficiency ηLC of linear-to-circular polarization conversion when light is reflected at a dielectric–conductor interface is determined as a function of the principal angle ϕ¯ and principal azimuth ψ¯ . Constant- ηLC contours are presented in the ϕ¯ ,ψ¯ plane for values of ηLC from 0.5 to 1.0 in steps of 0.05, and the corresponding contours in the complex plane of the relative dielectric function ϵ are also determined. As specific examples, efficiencies ⩾88% are obtained for light reflection by a Ag mirror in the visible and near-IR (400–1200nm) spectral range, and ≥40% for the reflection of extreme ultraviolet (EUV) and soft x-ray radiation by a SiC mirror in the 60–120nm wavelength range.

Embedded centrosymmetric multilayer stacks as complete-transmission quarter-wave and half-wave retarders under conditions of frustrated total internal reflection

S. R. Perla et al. — Fri, 11 May 2012 13:38:46 PDT

A centrosymmetric multilayer stack of two transparent materials, which is embedded in a high-index prism, can function as a complete-transmission quarter-wave or half-wave retarder (QWR or HWR) under conditions of frustrated total internal reflection. The multilayer consists of a high-index center layer sandwiched between two identical low-index films with high-index–low-index bilayers repeated on both sides of the central trilayer, maintaining the symmetry of the entire stack and constituting a QWR ( Δt=90° or 270°) or HWR (Δt=180°) in transmission. A QWR design at wavelengthλ=1.55 μm is presented that employs an 11-layer stack of Si and SiO2 thin films, which is embedded in a GaP cube prism. The intensity transmittances for the p and s polarizations remain >99% and Δt deviates from 90° by <±3°over a 100 nm spectral bandwidth (1.5⩽λ⩽1.6 μm) , and by ⩽±7° over an internal field view of ±1°(incidence angle 44°⩽ϕ0⩽46° inside the prism). An HWR design at λ=1.55 μm employs seven layers of Si andSiO2 thin films embedded in a Si cube, has an average transmittance >93% , and Δt that differs from 180° by<±0.3° over a 100 nm bandwidth (1.5⩽λ⩽1.6 μm) and by <±17° over an internal field view of ±1° . The sensitivity of these devices to film-thickness errors is also considered.

Quarter-wave layers with 50% reflectance for obliquely incident unpolarized light

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:45 PDT

The conditions under which light interference in a transparent quarter-wave layer of refractive index n1 on a transparent substrate of refractive index n2 leads to 50% reflectance for incident unpolarized light at an angle φ are determined. Two distinct solution branches are obtained that correspond to light reflection above and below the polarizing angle, φp , of zero reflection for p polarization. The real p and s amplitude reflection coefficients have the same (negative) sign for the solution branch φ>φp and have opposite signs for the solution branch φ<φp >. Operation at φ<φp is the basis of a>50%–50% beam splitter that divides an incident totally polarized light beam (with p and s components of equal intensity) into reflected and refracted beams of orthogonal polarizations [ Opt. Lett. 31, 1525 (2006) ] and requires a film refractive indexn1⩾(2√+1)n2−−√ . A monochromatic design that uses a high-index TiO2 thin film on a low-index MgF2substrate at 488 nm wavelength is presented as an example.

Phase shifts in frustrated total internal reflection and optical tunneling by an embedded low-index thin film

R. M.A. Azzam — Fri, 11 May 2012 13:38:44 PDT

Simple and explicit expressions for the phase shifts that p- and s-polarized light experience in frustrated total internal reflection (FTIR) and optical tunneling by an embedded low-index thin film are obtained. The differential phase shifts in reflection and transmission Δr,Δt are found to be identical, and the associated ellipsometric parameters ψr,ψt are governed by a simple relation, independent of film thickness. When the Fresnel interface reflection phase shifts for the pand s polarizations or their average are quarter-wave, the corresponding overall reflection phase shifts introduced by the embedded layer are also quarter-wave for all values of film thickness. In the limit of zero film thickness (i.e., for an ultrathin embedded layer), the reflection phase shifts are also quarter-wave independent of polarization (p or s) or angle of incidence (except at grazing incidence). Finally, variable-angle FTIR ellipsometry is shown to be a sensitive technique for measuring the thickness of thin uniform air gaps between transparent bulk media.

Achromatic angle-insensitive infrared quarter-wave retarder based on total internal reflection at the Si–SiO2 interface

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:42 PDT

An achromatic infrared (λ = 1.2–4 μm), Si-prism quarter-wave retarder (QWR) is described that uses total internal reflection at a buried Si–SiO₂ interface at an angle of incidence φ near 33°, where ∂Δ/∂φ = 0. The retardance Δ deviates from 90° by <±2° within a field of view of ±10° (in air) over the entire bandwidth. Because the SiO₂ layer at the base of the prism is optically thick, this QWR is unaffected by environmental contamination.

Phase shifts that accompany total internal reflection at a dielectric–dielectric interface

R. M.A. Azzam — Fri, 11 May 2012 13:38:42 PDT

The absolute, average, and differential phase shifts that p- and s-polarized light experience in total internal reflection (TIR) at the planar interface between two transparent media are considered as functions of the angle of incidence φ. Special angles at which quarter-wave phase shifts are achieved are determined as functions of the relative refractive index N. When the average phase shift equals π/2, the differential reflection phase shift Δ is maximum, and the reflection Jones matrix assumes a simple form. For N>√3, the average and differential phase shifts are equal (hence δ_p=3δ_s) at a certain angle φ that is determined as a function of N. All phase shifts rise with infinite slope at the critical angle. The limiting slope of the Δ-versus-φ curve at grazing incidence (∂Δ/∂φ)_φ=90°=−(2/N)(N²−1)^1/2=−2 cos φ_c, where φ_c is the critical angle and (∂²Δ/∂φ²)_φ=90°=0. Therefore Δ is proportional to the grazing incidence angle θ=90°−φ (for small θ) with a slope that depends onN. The largest separation between the angle of maximum Δ and the critical angle is 9.88° and occurs when N=1.55377. Finally, several techniques are presented for determining the relative refractive index N by using TIR ellipsometry.

Optimal beam splitters for the division-of-amplitude photopolarimeter

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:41 PDT

Optimal optical parameters of the beam splitter that is used in the division-of-amplitude photopolarimeter are determined. These are (1) 50%–50% split ratio of the all-dielectric beam splitter, (2) differential phase shifts in reflection and transmission Δ_r and Δ_t that differ by ±π/2, and (3) ellipsometric parameters (ψ_r, ψ_t)= (27.368°, 62.632°) or (62.632°, 27.368°). It is also shown that for any nonabsorbing beam splitter that splits incident unpolarized light equally, the relationship ψ_r+ψ_t=π/2 is always satisfied.

Angular range for reflection of p-polarized light at the surface of an absorbing medium with reflectance below that at normal incidence

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:39 PDT

The range of incidence angle, 0 < φ < φ_e, over which p-polarized light is reflected at interfaces between transparent and absorbing media with reflectance below that at normal incidence is determined. Contours of constant φ_e in the complex plane of the relative dielectric constant ε are presented. A method for determining the real and imaginary parts of the complex refractive index, ε^1/2 = n + jk, which is based on measuring φ_e and the pseudo-Brewster angle φ_pB, is viable in the domain of fractional optical constants, n, k < 1.

Tilted bilayer membranes as simple transmission quarter-wave retardation plates

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:38 PDT

A tilted bilayer membrane, which consists of two thin films of transparent optically isotropic materials of different refractive indices, can function as a transmission quarter-wave retarder (QWR) at a high angle of incidence. A specific design using a cryolite-Si membrane in the infrared is presented, and its tolerances to small shifts of wavelength, incidence angle, and film thickness errors are discussed. Some designs provide a dual QWR in transmission and reflection. Such devices provide simple linear-to-circular (and circular-to-linear) polarization transformers. Bilayer eighth-wave retarders without diattenuation are also introduced.

Poincaré sphere representation of the fixed-polarizer rotating-retarder optical system

R. M.A. Azzam — Fri, 11 May 2012 13:38:37 PDT

The trajectory of the polarization state of a monochromatic light beam after it passes through a fixed linear polarizer and a rotating linear retarder of arbitrary retardance Δ is determined on the Poincaré sphere. The three-dimensional figure-8 contour is shown to be the line of intersection of a right-circular cylinder with the sphere. The cylinder is parallel to the polar (s₃) axis, touches the sphere at the equator (at the point that represents the linear polarization transmitted by the fixed polarizer), and has a radius r=sin²(Δ/2). Projections of the trajectory in the coordinate planes of the normalized Stokes parameter space (s₁, s₂, s₃) are also determined.

Differential reflection phase shift under conditions of attenuated internal reflection

R. M.A. Azzam — Fri, 11 May 2012 13:38:36 PDT

The angle-of-incidence dependence of the differential reflection phase shift Δ between p and s polarizations is considered a function of the real and imaginary parts of the relative complex dielectric function ε of an interface in the domain of fractional optical constants, i.e., under conditions of internal reflection. The constraint on complex ε such that oscillatory and monotonic angular responses are obtained is determined. A sensitive and stable technique, which is based on attenuated internal reflection ellipsometry between the Brewster angle and the critical angle, is proposed for measuring small induced absorption (ε_i∼10⁻⁵) in the medium of refraction.

Single-layer-coated surfaces with linearized reflectance versus angle of incidence: application to passive and active silicon rotation sensors

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:35 PDT

A transparent or absorbing substrate can be coated with a transparent thin film to produce a linear reflectanceversus- angle-of-incidence response over a certain range of angles. Linearization at and near normal incidence is a special case that leads to a maximally flat response for p-polarized, s-polarized, or unpolarized light. For midrange and high-range linearization with moderate and high slopes, respectively, the best results are obtained when the incident light is s polarized. Application to a Si substrate that is coated with a SiO₂ film leads to novel passive and active reflection rotation sensors. Experimental results and an error analysis of this rotation sensor are presented.

Direct relation between Fresnel's interface reflection coefficients for the parallel and perpendicular polarizations: erratum 2

R. M.A. Azzam — Fri, 11 May 2012 13:38:34 PDT

The record is set straight concerning two equations that determine the reflection phase shifts at a single interface from the intensity reflectances forp- and s-polarized light at one angle of incidence. These equations appeared previously in this journal [J. Opt. Soc. Am. 69, 1007 (1979); erratum, J. Opt. Soc. Am. 70, 261 (1980)].

Photopolarimeter based on planar grating diffraction

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:33 PDT

A division-of-amplitude photopolarimeter (DOAP) is described that employs a diffraction grating in the conventional spectrometer orientation with the grating grooves normal to the plane of incidence. Four coplanar diffracted orders are used for polarimetric analysis to determine all four Stokes parameters of incident light simultaneously and virtually instantaneously (with the speed being determined solely by the photodetectors and their associated electronics); a fifth order is used for alignment by autocollimation or by use of a position-sensing quadrant detector. To sensitize the instrument for the +45° and -45° azimuths of incident linearly polarized light and for the handedness of incident circular polarization (i.e., for the third and fourth Stokes parameters), we insert two linear polarizers in two diffracted orders with their transmission axes inclined at appropriate angles with respect to the plane of incidence. The calibration and testing of an instrument of this type that uses an Al-coated 600-groove/mm holographic grating at 632.8-nm wavelength are reported as an example.

Limaçon of Pascal locus of the complex refractive indices of interfaces with maximally flat reflectance-versus-angle curves for incident unpolarized light

R. M.A. Azzam — Fri, 11 May 2012 13:38:32 PDT

For an interface between two isotropic media the power reflectance R_v (ø) is an even function, R_v (ø) = R_v (--), of the angle of incidence ø; hence all the odd derivatives, R_v⁽ⁿ⁾ = dⁿ R_v /døⁿ (n odd), are identically 0 at ø = 0, independent of the incident polarization v. When the incident light is unpolarized (v = u), the second derivative, R_u⁽²⁾, is also 0 at ø = 0, so that the flatness of the R_u -versus-ø curve over an initial range of ø starting from ø = 0 is determined by the fourth derivative,R_u⁽⁴⁾. The condition that R_u⁽⁴⁾ = 0 at ø = 0 gives the maximally flat response and leads to a specific constraint on the complex relative refractive index N, namely, that Re[(N - 1)²/N³] = 2/NN*. The corresponding complex plane contour is the limaçon of Pascal, η = 2 cos θ ± √3 in polar form, where N = η exp(jθ). The two branches of this contour constitute the boundary lines that separate the region of the complex plane in which the-function R_u (ø) is monotonic from that in which the function exhibits a minimum at oblique incidence. Families of curves that illustrate the maximally flat response in external and internal reflection are presented. New equations that determine the angle of incidence of minimum unpolarized-light reflectance of a dielectric-dielectric or a dielectric-conductor interface are derived.

Principal linear polarization states of an optical system

R. M.A. Azzam — Fri, 11 May 2012 13:38:31 PDT

The constraint on the Jones matrix of an optical system such that there exist two linear polarization states at its input that are mapped onto two corresponding linear states at its output is derived. These principal linear polarization (PLP) states, which characterize a broad range of systems, are also found in terms of the Jones matrix elements. Special cases when the PLP states are orthogonal, collapse onto one state, or become infinite in number are indicated. For a deterministic or nondeterministic optical system described by a Mueller matrix, the existence of two PLP states places a constraint on only 3 of the 16 matrix elements, namely, the first 3 elements of the last row. In general, the output light is partially linearly polarized. Several examples are given for demonstration.

Instrument matrix of the four-detector photopolarimeter: physical meaning of its rows and columns and constraints on its elements

R. M.A. Azzam — Fri, 11 May 2012 13:38:30 PDT

The four-detector photopolarimeter (FDP) is an arrangement of four photodetectors for measuring the state of polarization of light. The output current vector I of the FDP is related to the input Stokes vector S by I = AS, where A is the instrument matrix. The rows of A can be viewed as projection operators that determine the output currents of the detectors. This leads to the recognition of four special totally polarized input states, each of which maximizes the output of one detector. The associated four orthogonal states produce minimum signals. Because each detector is absorptive and its output is nonnegative, eight inequalities must be satisfied by the elements of A. For optically isotropic detectors, one element is identically zero and another can be made zero by an appropriate coordinate rotation. Three additional inequalities that are likely to apply are also stated. The columns of A have the following interesting meaning. The first column represents the normalized response of the FDP for incident unpolarized light or its average response to incident light polarized in any pair of orthogonal states. The second, third, and fourth columns represent the differential normalized responses of the FDP for incident light polarized in the following pairs of orthogonal states: (1) linear polarizations at 0 and 90° azimuths, (2) linear polarizations at 45° and 135° azimuths, and (3) the right and left circular polarizations, respectively. These are the same pairs of orthogonal states that are used in the phenomenological definitions of the Stokes parameters.

Accurate calibration of the four-detector photopolarimeter with imperfect polarizing optical elements

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:29 PDT

The first three columns of the instrument matrix A of the four-detector photopolarimeter (FDP) are determined by Fourier analysis of the output current vector I(P) as a function of the azimuth angle P of the incident linearly polarized light. Therefore 12 of the 16 elements of A are measured free of the imperfections of the (absent) quarter-wave retarder (QWR). The effect of angular beam deviation by the polarizer is compensated for by taking the average, (1/2) [I(P) + I(P + 180°)], of the FDP output at 180°-apart, optically equivalent, angular positions of the polarizer. The remaining fourth column of A is determined by the FDP’s response to the right- and left-handed circular polarization states. Because these states are impossible to generate with an imperfect QWR, a novel procedure is developed. In particular, the response of the FDP to the unattainable right- or left-handed circular polarization state is found by taking the average of the responses of the FDP to an elliptical near-circular state and that state rotated in azimuth by 90°. This calibration scheme is applied to measure A of our prototype FDP of four Si detectors at λ = 632.8 nm. A is determined, in external and internal reference frames, free of imperfections in the polarizing optical elements. The FDP, with its uncontaminated A matrix, is used subsequently to evaluate the imperfections of the QWR with the help of an appropriate model.

Analytical determination of the complex dielectric function of an absorbing medium from two angles of incidence of minimum parallel reflectance

R. M.A. Azzam — Fri, 11 May 2012 13:38:28 PDT

The real and imaginary parts of the complex dielectric function (or complex refractive index) of an opaque substrate or a thick film can be determined from two pseudo-Brewster angles measured in two transparent incidence media of different refractive indices. This two-angle method is simple in that it involves no photometric or polarimetric analysis and in that the solution for the optical properties in terms of the measured angles is explicit, analytical, and direct (i.e. noniterative). The two-angle method is demonstrated for an opaque TiN film on a Cleartran ZnS substrate as a specific example. The effect of angle-of-incidence errors on the determination of the optical properties is investigated, and the domain of applicability of this new and interesting method is also delineated.

Thin-film devices for polarized light- introduction

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:27 PDT

Division-of-wave-front thin-film beam splitter for generating binary patterns of orthogonal elliptical polarization states

R. M.A. Azzam — Fri, 11 May 2012 13:38:26 PDT

A division-of-wave-front thin-film beam splitter is described that reflects monochromatic light at oblique incidence with orthogonal elliptical polarization states. It consists of a metallic substrate partially covered with a transparent thin film that inverts the ratio ρ of the complex p and s reflection coefficients at the principal angle of the metal. Any pattern of coated and uncoated areas of the substrate is imprinted upon the reflected wave front as a corresponding two-dimensional spatial binary polarization pattern. A specific design is given that uses a Au substrate at a wavelength of 632.8 nm. The effects of small errors in the film refractive index, the film thickness, and the angle of incidence are discussed. It is noted that a layer that inverts ρ at a certain (especially high) angle of incidence is an effective ρ-inverting layer at all angles.

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

R. M.A. Azzam — Fri, 11 May 2012 13:38:25 PDT

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 and 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, r_p (ø), and the ratio of complex reflection coefficients for the p and s polarizations, ρ(ø) = r_p (ø)/r_s (ø), are considered as functions of the angle of incidence ø. It is found that the same (cubic) equation that determines the pseudo-Brewster angle of minimum |r_p | also determines a new angle at which the reflection phase shift δ_p = arg r_p exhibits 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.

General analysis and optimization of the four-detector photopolarimeter

R. M.A. Azzam et al. — Fri, 11 May 2012 13:38:24 PDT

The four-detector photopolarimeter (FDP) is analyzed for an arbitrary spatial configuration and any reflection characteristics (r_i, ψ_i, Δ_i) of the first three detectors. The instrument matrix A, which relates the output signal vector I to the input Stokes vector S by I = AS, and its determinant are derived explicitly. The essential condition that A be nonsingular (det A ≠ 0) is satisfied in general with uncoated absorbing detector surfaces, assuming that the plane of incidence (POI) is rotated between successive reflections by other than 90°. Therefore no special coatings on the detectors are required, and a thin dielectric (e.g., thermal oxide) layer would suffice. The differential reflection phase shift Δ is unrestricted for the first and 0third detectors and has optimum values of ±90° for the second. The optimum rotation angles of the POI are ±45° and ±135°. The optimum values of the surface parameter ψ are 27.37°, 22.5° or 67.5°, and 0 or 90° for the first, second, and third reflections, respectively. The following topics are also considered: (1) the partition of energy among detectors, (2) the effect of tilting the last detector, (3) operation of the FDP over a broadband spectral range, (4) choice of the light-beam path, and (5) calibration.

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

R. M.A. Azzam — Fri, 11 May 2012 13:38:22 PDT

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.

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

R. M.A. Azzam — Fri, 11 May 2012 13:38:21 PDT

The Fresnel reflection coefficients r_p and r_s of p- and s-polarized light at the planar interface between two linear isotropic media are found to be interrelated by (r_s - r_p)/(1 - r_sr_p) = 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), (r²_s - r_p)/(r_s - r_sr_p) = cos 2φ, which is valid at a given ø independent of ∊ (i.e., for all possible interfaces). Taken together, these two equations specify r_p and r_s completely and can be used to replace the original Fresnel equations.

Extinction of the p and s polarizations of a wave on reflection at the same angle from a transparent film on an absorbing substrate: applications to parallel-mirror crossed polarizers and a novel integrated polarimeter

R. M.A. Azzam — Fri, 11 May 2012 13:38:20 PDT

The p- and s-polarized components of light can be suppressed on reflection at the same angle of incidence from an absorbing substrate coated by a transparent thin film if the wave is refracted in the film at 45° and the constraint Re[(ε₂ - α)/(l -α)]^1/2 = α + | ε₂ - α| is satisfied, where 2α and ε₂ are the ratios of dielectric constants of the film and substrate, respectively, to that of the ambient. For high-reflectance metal substrates (|ε₂| » 1), α≈ 1, the ratio of film to ambient refractive index approaches √2, and the unextinguished reflectances approach 1. The least film thicknesses required to suppress the p and s polarizations are in the ratio 2:1. The analysis is applied to Si and Al substrates in the near UV-visible-near-IR spectral range. It is found that the film refractive index and thickness should be controlled to within ±0.01 and ±5 Å, respectively, for an A1 substrate at 550 nm. Significant applications are proposed that include parallel-mirror crossed polarizers, a novel polarimeter that integrates the polarization-analysis and photodetection functions, high-reflectance crossed thin-film reflection polarizers integrated on the same substrate, and division-of-wavefront polarizing beam splitters.

Explicit equations for the polarizing angles of a high-reflectance substrate coated by a transparent thin film

R. M.A. Azzam — Fri, 11 May 2012 13:38:19 PDT

Simple explicit equations are derived that determine the angles of incidence at which the parallel and perpendicular polarization components of light are extinguished on reflection from a transparent film coating a high-reflectance (metallic) substrate. The polarizing angles obtained from our approximate expressions are in excellent agreement with those determined by iterative numerical solution of the exact nonlinear equations that govern such angles. For the approximation to be valid, the intensity reflectance of the film-substrate interface, evaluated at the critical angle of the film-ambient interface, must exceed 0.5.

Inverting the ratio of the complex parallel and perpendicular reflection coefficients of an absorbing substrate using a transparent thin-film coating

R. M.A. Azzam — Fri, 11 May 2012 13:38:18 PDT

An absorbing substrate can be coated with a transparent thin film of refractive index N₁ (within a certain range) and thickness d such that the ratio of complex reflection coefficients for the p_and s polarizations of the film-covered substrate ρ = R_p/R_s is the inverse of that of the film-free substrate ρ¯ = R¯_p/R¯_s at an angle of incidence ø. A method to determine the relationship among ø, N₁, and d that inverts ρ (i.e., makes ρ = 1/ρ¯) for a given substrate at a given wavelength is described and is applied to aluminum and silver substrates at 0.6328- and 10.6-μm wavelengths, respectively. Sensitivity of the inversion condition to incidence-angle and film-thickness errors is analyzed. ρ-inverting layers can be applied to one of the two metallic mirrors of a beam displacer or axicon to preserve the polarization state of incident monochromatic radiation.

Keynote Speaker - Generational Crossroads: When X, Y, & Boomers Collide!

Eric Rowles — Thu, 19 Apr 2012 12:15:00 PDT

What do each of these generations have in common? And just as important, what makes them so different—in the workplace, in the community, in their day-to-day lives? This POWERFUL presentation helps deepen the connection and communication across the ages.