Photoreflectance/scattering measurements of spider silks informed by standard optics.

The silks of certain orb weaving spiders are emerging as high-quality optical materials. This motivates study of the optical properties of such silk and particularly the comparative optical properties of the silks of different species. Any differences in optical properties may impart biological advantage for a spider species and make the silks interesting for biomimetic prospecting as optical materials. A prior study of the reflectance of spider silks from 18 species reported results for three species of modern orb weaving spiders (Nephila clavipes, Argiope argentata and Micrathena Schreibersi) as having reduced reflectance in the UV range. (Modern in the context used here means more recently derived.) The reduced UV reflectance was interpreted as an adaptive advantage in making the silks less visible to insects. Herein, a standard, experimental technique for measuring the reflectance spectrum of diffuse surfaces, using commercially available equipment, has been applied to samples of the silks of four modern species of orb weaving spiders: Phonognatha graeffei, Eriophora transmarina, Nephila plumipes and Argiope keyserlingi. This is a different technique than used in the previous study. Three of the four silks measured have a reduced signal in the UV. By taking the form of the silks as optical elements into account, it is shown that this is attributable to a combination of wavelength-dependent absorption and scattering by the silks rather than differences in reflectance for the different silks. Phonognatha graeffei dragline silk emerges as a very interesting spider silk with a flat 'reflectance'/scattering spectrum which may indicate it is a low UV absorbing dielectric micro-fibre. Overall the measurement emerges as having the potential to compare the large numbers of silks from different species to prospect for those which have desirable optical properties.

Insights on correlation dimension from dynamics mapping of three experimental nonlinear laser systems.

BACKGROUND: We have analysed large data sets consisting of tens of thousands of time series from three Type B laser systems: a semiconductor laser in a photonic integrated chip, a semiconductor laser subject to optical feedback from a long free-space-external-cavity, and a solid-state laser subject to optical injection from a master laser. The lasers can deliver either constant, periodic, pulsed, or chaotic outputs when parameters such as the injection current and the level of external perturbation are varied. The systems represent examples of experimental nonlinear systems more generally and cover a broad range of complexity including systematically varying complexity in some regions. METHODS: In this work we have introduced a new procedure for semi-automatically interrogating experimental laser system output power time series to calculate the correlation dimension (CD) using the commonly adopted Grassberger-Proccacia algorithm. The new CD procedure is called the 'minimum gradient detection algorithm'. A value of minimum gradient is returned for all time series in a data set. In some cases this can be identified as a CD, with uncertainty. FINDINGS: Applying the new 'minimum gradient detection algorithm' CD procedure, we obtained robust measurements of the correlation dimension for many of the time series measured from each laser system. By mapping the results across an extended parameter space for operation of each laser system, we were able to confidently identify regions of low CD (CD < 3) and assign these robust values for the correlation dimension. However, in all three laser systems, we were not able to measure the correlation dimension at all parts of the parameter space. Nevertheless, by mapping the staged progress of the algorithm, we were able to broadly classify the dynamical output of the lasers at all parts of their respective parameter spaces. For two of the laser systems this included displaying regions of high-complexity chaos and dynamic noise. These high-complexity regions are differentiated from regions where the time series are dominated by technical noise. This is the first time such differentiation has been achieved using a CD analysis approach. CONCLUSIONS: More can be known of the CD for a system when it is interrogated in a mapping context, than from calculations using isolated time series. This has been shown for three laser systems and the approach is expected to be useful in other areas of nonlinear science where large data sets are available and need to be semi-automatically analysed to provide real dimensional information about the complex dynamics. The CD/minimum gradient algorithm measure provides additional information that complements other measures of complexity and relative complexity, such as the permutation entropy; and conventional physical measurements.

Variance of permutation entropy and the influence of ordinal pattern selection.

Permutation entropy (PE) is a widely used measure for complexity, often used to distinguish between complex systems (or complex systems in different states). Here, the PE variance for a stationary time series is derived, and the influence of ordinal pattern selection, specifically whether the ordinal patterns are permitted to overlap or not, is examined. It was found that permitting ordinal patterns to overlap reduces the PE variance, improving the ability of this statistic to distinguish between complex system states for both numeric (fractional Gaussian noise) and experimental (semiconductor laser with optical feedback) systems. However, with overlapping ordinal patterns, the precision to which the PE variance can be estimated becomes diminished, which can manifest as increased incidences of false positive and false negative errors when applying PE to statistical inference problems.

Permutation entropy with vector embedding delays.

Permutation entropy (PE) is a statistic used widely for the detection of structure within a time series. Embedding delay times at which the PE is reduced are characteristic timescales for which such structure exists. Here, a generalized scheme is investigated where embedding delays are represented by vectors rather than scalars, permitting PE to be calculated over a (D-1)-dimensional space, where D is the embedding dimension. This scheme is applied to numerically generated noise, sine wave and logistic map series, and experimental data sets taken from a vertical-cavity surface emitting laser exhibiting temporally localized pulse structures within the round-trip time of the laser cavity. Results are visualized as PE maps as a function of embedding delay, with low PE values indicating combinations of embedding delays where correlation structure is present. It is demonstrated that vector embedding delays enable identification of structure that is ambiguous or masked, when the embedding delay is constrained to scalar form.

Permutation entropy of finite-length white-noise time series.

Permutation entropy (PE) is commonly used to discriminate complex structure from white noise in a time series. While the PE of white noise is well understood in the long time-series limit, analysis in the general case is currently lacking. Here the expectation value and variance of white-noise PE are derived as functions of the number of ordinal pattern trials, N, and the embedding dimension, D. It is demonstrated that the probability distribution of the white-noise PE converges to a χ^{2} distribution with D!-1 degrees of freedom as N becomes large. It is further demonstrated that the PE variance for an arbitrary time series can be estimated as the variance of a related metric, the Kullback-Leibler entropy (KLE), allowing the qualitative N≫D! condition to be recast as a quantitative estimate of the N required to achieve a desired PE calculation precision. Application of this theory to statistical inference is demonstrated in the case of an experimentally obtained noise series, where the probability of obtaining the observed PE value was calculated assuming a white-noise time series. Standard statistical inference can be used to draw conclusions whether the white-noise null hypothesis can be accepted or rejected. This methodology can be applied to other null hypotheses, such as discriminating whether two time series are generated from different complex system states.

An exact surface-integral approach for accurate interferometric microscopy of single nanoparticles.

We present a half-plane surface-integral equation (SIE) approach for modeling the optical phase response of a single nanowire under phase-stepping interferometric (PSI) microscopy. This approach calculates scattered fields exactly from the Helmholtz equation in this 2D problem, obviating the need for ray-optic approximations. It is demonstrated that refractive index metrology is enabled by this method, with precision as low as 7 × 10(-5) possible for current state-of-the-art PSI microscopes. For nanowires of known refractive index, radii as small as 0.001λ are shown to yield a measurable phase signal and are therefore potentially measurable by this approach. Measurements are also demonstrated to be relatively insensitive to the spectral and coherence characteristics of the light source, the illumination conditions, and variations in nanowire cross-section shape. Prospects for measuring both the radius and refractive index simultaneously, and scope for generalizing this approach to arbitrary nanoparticle shapes are discussed.

Measuring nanoparticle size using phase-stepping interferometry: quantifying measurement sensitivity to surface roughness.

A method for sizing nanoparticles using phase-stepping interferometry has been developed recently by Little et al. [Appl. Phys. Lett. 103, 161107 (2013)]. We present an analytical procedure to quantify how sensitive measurement precision is to surface roughness. This procedure computes the standard deviation in the measured phase as a function of the surface roughness power spectrum. It is applied to nanospheres and nanowires on a flat plane and also a flat plane in isolation. Calculated sensitivity levels demonstrate that surface roughness is unlikely to be the limiting factor on measurement precision when measuring nanoparticle size using this phase-shifting-interferometry-based technique. The need to use an underlying surface that is very smooth when measuring nanoparticles is highlighted by the analysis.

Measuring nanoparticle size using optical surface profilers.

Optical surface profilers are state-of-the-art instruments for measuring surface height profiles. They are not conventionally applied to nanoparticle measurements due to the presence of diffraction artifacts. Here we use a theoretical model based on wave-optics to account for diffraction-based artifacts in optical surface profilers. This then enables accurate measurement of nanoparticles size of a known geometry. The model is developed for both phase shifting interferometry and vertical scanning interferometry modes of operation. It is demonstrated that nanosphere radii as small as 12 nm, and nano-cylinder radii as small as 10-15 nm can be measured from a standard profile measurement using phase shifted interferometry interpreted using the wave-optics approach.