<title>Refractive index evaluation using reflectometry: accuracy analysis</title>

Refractive index is one of the most important physical parameters of the materials. Owe to its great influence on the working characteristics of an optical systems, high accuracy measurement is required. Many methods have been proposed such as the v-prism method, the minimum deviation angle method, and the interferometric method. However, for the restriction of the principle, the shape of the sample is required to be a parallel plate or a prism with a specific shape. The sample with only spherical or aspheric surfaces cannot be tested. In this paper, an improved Brewster method is proposed to measure the refractive index of optical materials with arbitrary shapes. Brewster law can be expressed as that the reflectivity of the P-polarized light approaches zero when it is incident in the Brewster angle, which is the inverse trigonometric value of the refractive index. In the original method, a parallel laser beam with P polarization is incident on the sample, and reflected by it to a photodiode to get the intensity. The minimum intensity position corresponding to Brewster angle can be found by changing the incident angle. The reflecting surface of the sample need to be planar to provide smooth reflective area with a size greater than the beam diameter. In the improved method, a laser probe focusing on the sample and an array detector are used instead of the parallel beam and the photodiode. The minimum intensity position can be found with image processing technology. Since the laser beam is focused on the sample, only a tiny area with a size of 10 microns for reflection is needed. Thus, whatever the shape of the sample is, the method can be used. In order to demonstrate the feasibility, samples with different shapes such as a prism, a parallel plate and a lens was tested, and the accuracy of the results could all reach the order of magnitude of 10-3.

This article presents a three autocollimation method (TAM) for measuring the refractive index (RI) of optical glass. The popular minimum deviation method (MDM) for RI measurement is accurate, but it is very laborious, since the procedure of determining the minimum possible angle requires many iterations. TAM allows the calculation of RI based on the measurement of the angles of incidence of the beam corresponding to the autocollimation from the mirror located behind the investigated prism, parallel to the entrance face. Measurements must be carried out for all three faces, but it is not necessary to measure the prism angles, since the RI is calculated by solving a system of equations relating the values of unknown parameters. A comparative analysis showed that the accuracy of the proposed method is about 20% higher than the accuracy of classical MDM under the same conditions. A limitation of TAM is the phenomenon of total internal reflection, which limits the measurement range. To implement TAM, a goniometric system with a photoelectric encoder and a set of three reference prisms made of optical glass is used. The experimental absolute error does not exceed ±1·10 −5 , which proves the prospect of using the method for high‐precision RI measurements. TAM can also be used to study the dispersion characteristics of triangular prisms made of optically transparent materials. TAM increases the accuracy and simplifies the RI measurement process compared to MDM.

Thermodilution cardiac output using a pulmonary artery catheter is the reference method against which all new methods of cardiac output measurement are judged. However, thermodilution lacks precision and has a quoted precision error of ± 20%. There is uncertainty about its true precision and this causes difficulty when validating new cardiac output technology. Our aim in this investigation was to determine the current precision error of thermodilution measurements. A test rig through which water circulated at different constant rates with ports to insert catheters into a flow chamber was assembled. Flow rate was measured by an externally placed transonic flowprobe and meter. The meter was calibrated by timed filling of a cylinder. Arrow and Edwards 7Fr thermodilution catheters, connected to a Siemens SC9000 cardiac output monitor, were tested. Thermodilution readings were made by injecting 5 mL of ice-cold water. Precision error was divided into random and systematic components, which were determined separately. Between-readings (random) variability was determined for each catheter by taking sets of 10 readings at different flow rates. Coefficient of variation (CV) was calculated for each set and averaged. Between-catheter systems (systematic) variability was derived by plotting calibration lines for sets of catheters. Slopes were used to estimate the systematic component. Performances of 3 cardiac output monitors were compared: Siemens SC9000, Siemens Sirecust 1261, and Philips MP50. Five Arrow and 5 Edwards catheters were tested using the Siemens SC9000 monitor. Flow rates between 0.7 and 7.0 L/min were studied. The CV (random error) for Arrow was 5.4% and for Edwards was 4.8%. The random precision error was ± 10.0% (95% confidence limits). CV (systematic error) was 5.8% and 6.0%, respectively. The systematic precision error was ± 11.6%. The total precision error of a single thermodilution reading was ± 15.3% and ± 13.0% for triplicate readings. Precision error increased by 45% when using the Sirecust monitor and 100% when using the Philips monitor. In vitro testing of pulmonary artery catheters enabled us to measure both the random and systematic error components of thermodilution cardiac output measurement, and thus calculate the precision error. Using the Siemens monitor, we established a precision error of ± 15.3% for single and ± 13.0% for triplicate reading, which was similar to the previous estimate of ± 20%. However, this precision error was significantly worsened by using the Sirecust and Philips monitors. Clinicians should recognize that the precision error of thermodilution cardiac output is dependent on the selection of catheter and monitor model.

A simple method to measure the refractive index of a glass prism with very low uncertainty was developed at INRiM. The method is a modification of the classical minimum deviation method. A brief description of the methods used to measure the vertex angles of the prism and the angle of minimum deviation is reported together with the uncertainty evaluation. The technique is going to be validated by a comparison between INRiM and two other laboratories. A relative standard uncertainty better than 1 ppm has been obtained.

The theory, and the use at normal incidence, of shear-vertically polarized waves (with polarization vector in the plane containing the incident wave vector and the normal on the interface) using the mode conversion method has been tackled by others. Here we develop the theory for shear-horizontally polarized incident waves (with polarization vector perpendicular to both the normal on the interface and the incoming wave vector). We take into account normal incidence as well as oblique incidence. For normal incidence, we discover the generation of Love waves. If oblique incidence is considered, we discover the existence of a Brewster angle of incidence, comparable with the Brewster angle in optics, in which a diffraction grating can be used as a polarization filter.

The pairs of observations at non-normal incidence which specify the various reflection methods of determining the optical constants n and k are compared as to their arithmetical sensitivity to changes in n and k and hence as to their theoretical suitability. Only the simple methods which do not require polarimetric analysis are considered. The comparison of sensitivity is done by inspection of charts prepared from computed tables and which cover a wide range of n and k. A previously unnoticed property of the Fresnel equations at 45° incidence allows any method, not just one in which a ratio of reflectivities is measured, to be employed without a standard reflector. Experimental factors as well as theoretical sensitivity are considered, and the use of the Brewster angle as one observation is proposed. An exact analysis of the Brewster condition has been made, which leads to explicit equations applicable to two proposed methods. The distinction between the Brewster angle and the principal angle of incidence, which are often confused, is emphasized.

The experience of developing and applying methods for measuring the refractive coefficients of crystals of the highest and middle categories based on multi-angle spectrophotometric reflection methods is presented: by the reflection spectrum from one face at an angle of incidence of light close to normal, and by the reflection method when light falls at Brewster angle. The advantages and limitations of the methods and the requirements for the samples are described. It is shown that the reflection method at an angle of incidence close to normal is applicable for optically isotropic media. The Brewster angle method is applicable for crystals of the highest and middle categories. The measurement accuracy of both methods has been determined. The applicability of these methods is shown for samples of crystals of the highest and middle categories Gd3Al2Ga3O12:Ce and La3Ga5.5Ta0.5O14 respectively.

We summarize the results of our comprehensive analytical and numerical studies of the effects of polarization on the Anderson localization of classical waves in one-dimensional random stacks. We consider homogeneous stacks composed entirely of normal materials or metamaterials, and also mixed stacks composed of alternating layers of a normal material and a metamaterial. We extend the theoretical study developed earlier for the case of normal incidence to the case of off-axis incidence. For the general case where both the refractive indices and layer thicknesses are random, we obtain the long-wave and short-wave asymptotics of the localization length over a wide range of incidence angles (including the Brewster "anomaly" angle). At the Brewster angle, we show that the long-wave localization length is proportional to the square of the wavelength, as for the case of normal incidence, but with a proportionality coefficient substantially larger than that for normal incidence. In mixed stacks with only refractive-index disorder, we demonstrate that p -polarized waves are strongly localized, while for s polarization the localization is substantially suppressed, as in the case of normal incidence. In the case of only thickness disorder, we study also the transition from localization to delocalization at the Brewster angle.

Reflectivity characteristics of the crust from modelling refraction/wide-angle and normal-incidence reflection data on BABEL Line 1

Liquid Mercury (Hg)

In the year 1620 the printing office of the University of Prague published a 58page table containing the values an = (1.0001) n for 0 ≤ n ≤ 23027, rounded to 9 decimal digits. This table had been devised and computed about 20 years earlier by the Swiss-born astronomer and watchmaker Jost Burgi in order to facilitate the multi-digit multiplications and divisions he needed for his astronomical computations. The “Progres Tabulen”, as Burgi called his tables, are considered to be one of the two independent appearances of the logarithms in the history of mathematics the other one, due to John Napier (1550-1617), appeared in 1614. There are only a few copies of the original printing extant: one of them is now in the Astronomisch-Physikalisches Kabinett in Munich. Based on a copy of this original, the terminal digits of all table entries were extracted and compared with the exact values of an, a matter of a split second on a modern computer. In this presentation we give a brief account of the mathematical environment at the end of the 16th century as well as a detailed description of Burgi’s Progres Tabulen and their application to numerical computations. We will also give a sketch of Burgi’s remarkable life and of his numerous achievements besides the discovery of the logarithms. Our main purpose, however, is to analyze the numerical errors in Burgi’s table. First of all, there are no systematic errors, e.g. the crux of the table, 1.0001 = 10, is correct with all digits given. 91.5% of the table entries are entirely correct, and 7.3% of the values show round-off errors between 0.5 and 1 unit of the least significant digit. The remaining 1.17% table errors are mainly errors of transcription and illegible digits. Statistics of the round-off errors leads to interesting conclusions concerning Burgi’s algorithms of generating his table and on his handling of the round-off errors, as well as on the computational effort involved.

We investigate an interdigitated nanograting structure on a GaAs substrate for plasmonic metal–semiconductor–metal photodetector applications. This computational work has studied the effects that the taper angle of the nanograting sidewall and the light wave angle of incidence have on the optical and current enhancements in the device. The study, involving two types of taper angle structures—positive and negative—showed that both taper angle directions can generate more optical and electrical enhancements than perfectly vertical wall structures for light incident at both the Brewster angle and the normal incident angle. The enhanced electric field value at the optimum positive taper angle is ∼22% and ∼120% greater than the negative taper angle and vertical wall structure, respectively. In addition, the total weighted optical enhancement value for the optimal positive taper angle structure is ∼65% and ∼120% greater than the optimal negative taper angle and vertical wall structure, respectively. This work demonstrates that the increased enhancements are due to the nanoscale focusing of light and impedance matching. The incident wave angle along with the taper angle can significantly promote these enhancements, especially at the Brewster angle.

Polarity reversal is a well-known problem in elastic reverse time migration, and it is closely related to the imaging conditions. The dot product of source and receiver wavefields is a stable and efficient way to construct scalar imaging conditions for decomposed elastic vector wavefields. However, for PP images, the dot product introduces an angle-dependent factor that will change the polarity of image amplitudes at large opening angles, and it is also contaminated by low-wavenumber artifacts when sharp contrasts exist in the velocity model. Those two problems can be suppressed by muting the reflections with large opening angles at the expense of losing useful information. We have developed an elastic inverse-scattering imaging condition that can retain the initial polarity of the image amplitude and significantly reduce the low-wavenumber noise. For PS images, much attention is paid to the polarity-reversal problem at the normal incidence, and the dot-product-based imaging condition successfully avoids this kind of polarity reversal. There is another polarity-reversal problem arising from the sign change of the PS reflection coefficient at the Brewster angle. However, this sign change is often neglected in the construction of a stacked PS image, which will lead to reversed or distorted phases after stacking. We suggested using the S-wave impedance kernel used in elastic full-waveform inversion but only in the PS mode as an alternative to the dot-product imaging condition to alleviate this kind of polarity-reversal problem. In addition to dot-product-based imaging conditions, we analytically compare divergence- and curl-based imaging conditions and the elastic energy norm-based imaging condition with the presented imaging conditions to identify their advantages and weaknesses. Two numerical examples on a two-layer model and the SEAM 2D model are used to illustrate the effectiveness and advantages of the presented imaging conditions in suppressing low-wavenumber noise and correcting the polarity-reversal problem.

To date, most numerical and experimental studies of metamaterials are limited to normally incident plane waves with the magnetic field parallel to the axis of the inductive structure. In this article, the reflected field response of three types of Split Ring Resonator (SRR) media are measured as a function of incident field polarization, angle of incidence and frequency, using an automated free space focused beam measurement system. The measured results are compared with simulated data generated using the experimentally retrieved complex permittivity and permeability tensors from normal incidence measurements. It is concluded that there is no classical Brewster angle (zero reflection for a given angle of incidence). However, we observe a reflection minimum for all angles of incidence in a small range of frequencies close to the plasmon frequency for all orientations of the SRRs for perpendicular polarization. This will indeed have important low observable applications. In parallel polarization, SRR media are transparent for small angles of incidence. For both polarizations, the effect of anisotropy becomes significant as the angle of incidence increases and a significant change in the reflected field is observed. Thus, considering the anisotropic nature of such metamaterials is important when designing applications. In this article, we also independently verify the assumed constitutive models and property retrieval using normal incidence measurements, using oblique incidence scans. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 2619–2624, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21986

A zero-order rainbow angle may be defined as the relative minimum angle of deviation of geometrical light rays transmitted without internal reflections through a transparent particle. If the incident rays are parallel and the particle is a sphere, such a minimum does not exist. But if the incident rays are not parallel or if the particle has an elliptical rather than circular cross section, an angle of minimum deviation, hence a zero-order rainbow, can occur. For a spherical water droplet, the zero-order rainbow will occur when a point source is placed less than a droplet radius away from its surface. If a column of water with an elliptical cross section is illuminated by a plane wave, a zero-order rainbow will occur if the length of the major axis of the cross section is more than twice the length of the minor axis.