Tensors and matrices in optical mineralogy

1987 ◽  
Vol 51 (363) ◽  
pp. 655-663 ◽  
Author(s):  
A. Peckett

AbstractA series of matrix operations is described which enables the following optical data to be calculated for plane polarized light perpendicularly incident on a section of a transparent or opaque mineral: 1. vibration directions and refractive indices of anisotropic transparent minerals: 2. reflectivities and the state of polarization of light reflected from anisotropic opaque minerals. The data needed are the dielectric tensor, its orientation with respect to the crystal axes, the unit cell parameters and the direction of the incident light. The mathematical techniques involve the manipulation of matrices, the determination of eigenvalues and eigenvectors and, for opaque minerals, the manipulation of complex numbers. All operations can be carried out with the aid of some of the recent electronic calculators which have built-in matrix algebra procedures and complex arithmetic.

1989 ◽  
Vol 53 (369) ◽  
pp. 71-78 ◽  
Author(s):  
A. Peckett

AbstractAnisotropic opaque minerals viewed in reflected light microscopy show two sets of colours: the colours seen in plane polarized light which change as the section is rotated on the microscope stage, and the colours seen between crossed polars which change as the analyser is uncrossed. These latter colours are known variously as polarization colours or anisotropic rotation tints, but are here referred to as anisotropy colours. They are commonly a diagnostic aid to correct mineral identification. All these colours occur as a consequence of the dispersion of the relative permittivity (dielectric) tensor—the variation in the values of the tensor with wavelength of incident light and in low symmetry crystals, the variation in the directions of the principal axes of the tensor with wavelength.In this paper, it is shown that the colour seen in plane polarized light and the anisotropy colours can be predicted for any orientation of section, at any stage angle, and for any degree of uncrossing of the analyser by calculations based on the dielectric tensor values, and these predicted colours compare favourably with the observed values. Three minerals are studied in this paper as examples: stannite, covelline, and bournonite.


2020 ◽  
Vol 126 (9) ◽  
Author(s):  
Joachim Jelken ◽  
Carsten Henkel ◽  
Svetlana Santer

Abstract We study the peculiar response of photo-sensitive polymer films irradiated with a certain type of interference pattern where one interfering beam is S-polarized, while the second one is P-polarized. The polymer film, although in a glassy state, deforms following the local polarization distribution of the incident light, and a surface relief grating (SRG) appears whose period is half the optical one. All other types of interference patterns result in the matching of both periods. The topographical response is triggered by the alignment of photo-responsive azobenzene containing polymer side chains orthogonal to the local electrical field, resulting in a bulk birefringence grating (BBG). We investigate the process of dual grating formation (SRG and BBG) in a polymer film utilizing a dedicated set-up that combines probe beam diffraction and atomic force microscopy (AFM) measurements, and permits acquiring in situ and in real-time information about changes in local topography and birefringence. We find that the SRG maxima appear at the positions of linearly polarized light (tilted by 45° relative to the grating vector), causing the formation of the half-period topography. This permits to inscribe symmetric and asymmetric topography gratings with sub-wavelength period, while changing only slightly the polarization of one of the interfering beams. We demonstrate an easy generation of sawtooth profiles (blazed gratings) with adjustable shape. With these results, we have taken a significant step in understanding the photo-induced deformation of azo-polymer films.


1844 ◽  
Vol 15 (1) ◽  
pp. 37-65
Author(s):  
P. Kelland

The present Memoir is, to a certain extent, a continuation of one which the author presented to the Society in December 1838, and which has since been published in the thirteenth volume of the Transactions. Other motives, however, than the desire of completing the subject, have influenced him in producing the following analysis. A very important point in the hypothetical conditions which Fresnel assumed to hold with respect to polarized light, has, of late, been warmly combated in various quarters. Fresnel supposed that light polarized in a given plane consists of vibrations of such a nature that the motion is perpendicular to that plane. Neumann and other writers contend that the very opposite is the fact. We hope to be able to offer evidence of some little weight in favour of the former view; at the same time we do not pretend to shew the actual impossibility of the truth of the latter.


Author(s):  
B. G. Fitzpatrick ◽  
S. L. Keeling ◽  
S. G. Rock

Abstract A least squares reconstruction technique is examined for determining flow-field densities from optical data. Nonintrusive optical methods have long been used for flow visualization; however, the goal of this work is to devise mathematical techniques with which optical data can be used for quantitative flow measurement. The ill-posedness of density computation from interferogram measurements is recognized as a serious limitation in direct inversion methods. Here, least squares techniques employing compactness constraints are developed to avoid the difficulties encountered in traditional approaches.


This paper contains an experimental investigation of the phenomena of elliptic polarization resulting from the reflexion of polarized light from metallic surfaces, and the theory on which they are explicable; the analytical results being given in a tabular form, and applied to the cases of the experiments themselves.


2010 ◽  
Vol 17 (1) ◽  
pp. 125-131
Author(s):  
Howard J. Swatland

AbstractMicroscope and fiber-optic spectrophotometry of transmittance and backscattering both showed moss leaves to be capable of casting strong shadows, with a single leaf blocking approximately 90% of incident light from a point source. In leaves with only one layer of cells, the transmittance through the cytoplasm of single cells was similar to that for whole leaves. Analysis of cell wall birefringence by polarized-light interferometry indicated that cell walls might normally scatter rather than transmit light. Spectra transmitted through, or backscattered from, the upper green layers of moss were dominated by selective absorbance from chlorophyll, but there was also evidence of wavelength-dependent scattering, as detected in the lower layers of brown, dead moss. Specular reflectance from moss leaves was detected by polarimetry and may have contributed to the relatively high macroscopic transmittance of stationary moss in water. Shadowing by moss leaves was confirmed by dynamic measurements of mosses in turbulent water without bubbles. Flicker patterns from leaves were superimposed on the underwater flicker pattern created at the air-water interface, thus flecks of light were reduced in intensity, increased in frequency, and decreased in duration. This was detected with both point source and diffuse illumination of samples.


Optics f2f ◽  
2018 ◽  
pp. 51-70
Author(s):  
Charles S. Adams ◽  
Ifan G. Hughes

This chapter discusses the polarization of light, including the transverse nature of the plane-wave solution; the linear and circular bases are introduced, and the propagation of polarized light in media is analysed.


2019 ◽  
Vol 33 (25) ◽  
pp. 1950305 ◽  
Author(s):  
Wenhua Zhu ◽  
Bo Wang ◽  
Chenhao Gao ◽  
Kunhua Wen ◽  
Ziming Meng ◽  
...  

This paper designed a novel three-output reflective packaged grating. The optimized parameters such as the period and depth of the high-efficiency three-output grating with an incident wavelength of 1550 nm can be calculated by rigorous coupled-wave analysis (RCWA). According to the optimized result, the grating can diffract the incident light energy into three orders with an efficiency of nearly 33% under the premise of second Bragg angle incidence and the given duty ratio of 0.5. The diffraction efficiency of the packaged grating is improved compared to the surface-relief three-output grating under second Bragg angle incidence, especially for TE-polarized light.


The iridescent cuticle of certain Rutelino scarab beetles, which is a form optically active and selectively reflects circularly polarized light, incorporates an NH 4 OH -extractable component The ultraviolet absorption spectrum of this component, together with its chromatographic and refractive properties, identify it as uric acid (2,6,8-trihydroxypurine). All species of Plusiotis examined have uric acid in their reflecting layers, as do several species of Anoplognathus. Plusiotis resplendens has a reflecting layer with a uric acid volume fraction of 0.7, P . optima a volume fraction of 0.6. The reflecting layer of P . resplenden s has an anticlockwise helicoidal architecture, the optical thickness of the helicoidal p itch being such that it constructively interferes with visible light wavelengths. An anticlockwise helicoid constructively interferes with only the left circularly polarized component of incident light, right circularly polarized light being transmitted without attenuation. P. resplendens has a 1.8 /xm thick unidirectional layer embedded within the helicoid which functions as a perfect halfwave retardation plate for wavelength 590 nm . This halfwave plate enables the helicoidal reflector in this species to reflect both left and right circularly polarized components of incident light. After passing through the halfwave plate, transmitted right circularly polarized light becomes left circularly polarized ; this light is now reflected and emerges from the cuticle right circularly polarized, after passing back through the halfwave plate. Consequently the total reflectivity of circularly polarized incident light is greater in P. resplendens than in any other species examined; the plate also reduces multiple internal reflexions. Interferometric analysis of the refractive properties of the helicoidal reflectors in species of Plusiotis showed that the ordered incorporation of uric acid increases the birefringence of the system by a factor of five times, e.g. the in tact birefringence of the unidirectional layer of P . resplendens is 0.166 at wavelength 560 nm ; after uric acid extraction the birefringence is reduced to 0.034. As the coefficient of reflexion of a helicoidal reflector is directly proportional to the birefringence of the individual planes comprising the helicoid, beetles incorporating uric acid into their reflecting surfaces reflect circularly polarized light far more efficiently than beetles lacking uric acid. Refractive index values for a single multicomponent plane of the helicoid have been summarized as a biaxial indicatrix, with the Z axis tilte dat 45° to the plane of the epicuticle. Beetle reflecting layers which incorporate uric acid have twenty times greater optical rotatory power compared with reflecting layers lacking this component. Mathematical treatments dealing with helicoidal reflectors predict the form optical rotatory power to be a function of the square of the birefringence, which is in agreement with the experimental observations. To enable uric acid to have the optical effects mentioned above, an epitaxial incorporation into the helicoidal framework is necessary. Although uric acid is a common cytoplasmic reflecting material in arthropods, this is the first record of its presence in an extracellular (cuticular) reflector.


2020 ◽  
Vol 27 (09) ◽  
pp. 1950201
Author(s):  
CHEN FU ◽  
BO WANG ◽  
WENHUA ZHU ◽  
KUNHUA WEN ◽  
ZIMING MENG ◽  
...  

This paper designed a novel three-port reflective surface-covered grating with a connecting layer. The grating can be used as a splitter, and the polarized light can be divided into zero order, first order and second order. Through rigorous coupled-wave analysis, the efficiency of the three orders of diffraction light is close to 33% under the condition that the incident light at 1550 nm is incident at the second Bragg angle and the given duty cycle is 0.5. The efficiency and bandwidth of the surface-covered grating are improved compared with that of the surface-relief grating reported in the past. Especially for transverse magnetic polarized light, the beam splitting effect is more uniform, the efficiency ratio of the zeroth order to first order can reach 1.01, and the efficiency ratio of the first order to second order can reach 1.


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