A CAUTIONARY NOTE ON POLYNOMIAL DISTRIBUTED LAG FORMULATIONS OF SUPPLY RESPONSE

This paper uses the Pagano-Hartley procedure to estimate the lag length and polynomial degree for the case of a quarterly hog supply equation. The results show that the nicely humped shapes which materialize when using the Almon lag may be caused by the failure in accounting for autocorrelation in determining lag length and polynominal degree.Livestock Production/Industries, Research Methods/ Statistical Methods,

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

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\u3eφp and have opposite signs for the solution branch φ\u3cφp . Operation at φ\u3cφp 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

Reflection coefficients of p- and s-polarized light by a quarter-wave layer: explicit expressions and application to beam splitters

The complex-amplitude reflection coefficients of p- and s-polarized light by a transparent freestanding, embedded, or deposited quarter-wave layer (QWL) are derived as explicit functions of the angle of incidence and layer refractive index. This provides the basis for the design of 50%-50% beam splitters for incident s-polarized or unpolarized light that use a high-index (e.g., TiO2 or Ge) QWL embedded in a glass cube in the visible and near infrared spectral range. These simple devices have good angular and spectral response and are insensitive to small film thickness errors to the first order

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: −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

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 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 = 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-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

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

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

Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter

A design procedure is presented for a near-optimal, single-layer-coated prism beam splitter that serves as the key optical element of the division-of-amplitude photopolarimeter (DOAP). For given film and substrate refractive indices, the angle of incidence and film thickness are selected such that the ellipsometric differential phase shifts in reflection and transmission Delta_r and Delta_t differ by ±pi/2, and the normalized determinant of the instrument matrix is maximized. The best results are obtained by using high-index films on low-index substrates. This is illustrated by examples of ZnS and GaP films on silica prisms in the visible and Si, Ge, and PbTe films on Irtran 1 substrates in the infrared. A 16° Si-prism DOAP beam splitter at the 1.55-µm lightwave-communications wavelength is also presented. It uses a 163-nm SiO2 coating on the entrance face to satisfy the optimum delta condition at 73° incidence, and the determinant of the instrument matrix is 78.23% of its theoretical maximum. The exit face of the Si prism is antireflection coated with a 208-nm Si3N4 film

Single-layer-coated beam splitters for the division-of-amplitude photopolarimeter

A design procedure is presented for a near-optimal, single-layer-coated prism beam splitter that serves as the key optical element of the division-of-amplitude photopolarimeter (DOAP). For given film and substrate refractive indices, the angle of incidence and film thickness are selected such that the ellipsometric differential phase shifts in reflection and transmission Delta_r and Delta_t differ by ±pi/2, and the normalized determinant of the instrument matrix is maximized. The best results are obtained by using high-index films on low-index substrates. This is illustrated by examples of ZnS and GaP films on silica prisms in the visible and Si, Ge, and PbTe films on Irtran 1 substrates in the infrared. A 16° Si-prism DOAP beam splitter at the 1.55-µm lightwave-communications wavelength is also presented. It uses a 163-nm SiO2 coating on the entrance face to satisfy the optimum delta condition at 73° incidence, and the determinant of the instrument matrix is 78.23% of its theoretical maximum. The exit face of the Si prism is antireflection coated with a 208-nm Si3N4 film