Linear Phase Function Research Articles

Polygonal Laguerre-Gauss beams

We demonstrate that Laguerre-Gauss (LG) beams can have a polygonal distribution with the radial and azimuthal components, the so-called polygonal Laguerre-Gauss beams. These light beams are generated with a linear and trigonometric phase function of the angular coordinate which transform the Perfect LG beams with a lens. The polygonal Laguerre-Gauss beams preserve their distribution within a maximum propagation distance without diffraction similar to that used in the propagation-invariant light fields. We expect that these polygonal Laguerre-Gauss beams will be a method for creating any distribution controlling the radial and azimuthal components; moreover, they can be a proposal to control the number of the ring structures of the light beams.

Resolution-enhanced imaging by interferenceless coded aperture correlation holography with two competing methods of phase mask synthesis

Introducing a scattering phase mask between an object and an objective lens can improve the imaging resolution of a maskless system. The resolution enhancement is achieved using the technique of interferenceless coded aperture correlation holography with a point spread hologram (PSH) of N randomly distributed dots. For the same PSH type, there is more than one way to synthesize the phase mask. In this study, we suggest a new method of synthesizing masks and compare it with the previous method. The new method is random multiplexing of linear phase functions, each of which is responsible for a single dot on the camera plane. The previous method is based on the well-known iterative Gerchberg-Saxton algorithm with the constraints of a randomly chosen dot pattern on the camera plane and a pure phase function on the mask plane. In terms of the signal-to-noise ratio of the reconstructed image, the iterative algorithm is better for high N but worse for low N, as expected. This article has also answered the question of which method is preferred to synthesize the phase mask.

Characterizing the nucleus of comet 162P/Siding Spring using ground-based photometry

ABSTRACT Comet 162P/Siding Spring is a large Jupiter-family comet with extensive archival lightcurve data. We report new r-band nucleus light curves for this comet, acquired in 2018, 2021, and 2022. With the addition of these light curves, the phase angles, at which the nucleus has been observed, range from 0.39○ to 16.33○. We absolutely calibrate the comet light curves to r-band Pan-STARRS 1 magnitudes, and use these light curves to create a convex shape model of the nucleus by convex lightcurve inversion. The best-fitting shape model for 162P has axis ratios a/b = 1.56 and b/c = 2.33, sidereal period P = 32.864 ± 0.001 h, and a rotation pole oriented towards ecliptic longitude λE = 118○ ± 26○ and latitude βE = −50○ ± 21○. We constrain the possible nucleus elongation to lie within 1.4 < a/b < 2.0 and discuss tentative evidence that 162P may have a bilobed structure. Using the shape model to correct the light curves for rotational effects, we derive a linear phase function with slope β = 0.051 ± 0.002 mag deg−1 and intercept Hr(1, 1, 0) = 13.86 ± 0.02 for 162P. We find no evidence that the nucleus exhibited an opposition surge at phase angles down to 0.39°. The challenges associated with modelling the shapes of comet nuclei from light curves are highlighted, and we comment on the extent to which we anticipate that Legacy Survey of Space and Time will alleviate these challenges in the coming decade.

A comprehensive estimation method of phase error in the frequency‐interleaving digital‐to‐analog converter

This letter studied the phase error in the frequency-interleaving digital-to-analog converter (FI-DAC) and proposed a comprehensive phase estimation method. Firstly, the model of the FI-DAC system was established, and the phase error was considered that consists of three parts: time delay error, initial phase error and non-linear phase error. By analyzing the phase function characteristics of the system, the least squares (LS) was used to linearly fit the phase functions of the non-overlapping bands to estimate both the time delay and the initial phase of each sub-band channel. Then, the non-linear phase error of the system was estimated by the difference between the system phase function and the estimated linear phase function. Finally, the effectiveness of the proposed estimation method was verified by the built FI-DAC test bench.

Asteroid Photometric Phase Functions From Bayesian Lightcurve Inversion

Photometry is an important tool for characterizing the physical properties of asteroids. An asteroid’s photometric lightcurve and phase curve refer to the variation of the asteroid’s disk-integrated brightness in time and in phase angle (the Sun-asteroid-observer angle), respectively. They depend on the asteroid’s shape, rotation, and surface scattering properties, and the geometry of illumination and observation. We present Bayesian lightcurve inversion methods for the retrieval of the asteroid’s phase function, the unambiguous phase curve of a spherical object with surface scattering properties equal to those of the asteroid. A collection of such phase functions can give rise to a photometric taxonomy for asteroids. In the inverse problem, first, there are four classes of lightcurves that require individual error models. The photometric observations can be absolute or relative and they can have dense or sparse cadence in comparison to the rotation period of the asteroid. Second, the observations extend over varying phase angle ranges, requiring different phase function models. Asteroid photometry from the European Space Agency Gaia space mission extends, typically, over a range of phase angles, where the phase curve tends to be linear on the magnitude scale. Photometry from ground-based observing programs can reach small phase angles, where the asteroids show an opposition effect, a nonlinear increase of brightness on the magnitude scale towards zero phase angle. We provide error models for all four classes of lightcurves and make use of linear or linear-exponential phase functions for phase angles below 50°. We apply the inverse methods to sparse absolute Gaia and dense relative ground-based lightcurves and obtain absolute magnitudes and phase functions, with uncertainties, for ∼500 asteroids. Finally, we assess the lightcurve inversion problem for dense absolute photometry with the help of a numerical simulation for a Gaussian-random-sphere asteroid.

Discrete ordinates solution of the radiative transfer equation in media with strong forward and backward scattering subjected to collimated irradiation

Highly anisotropic phase functions are often approximated by simpler ones given by the sum of a Dirac delta function and a smooth function. However, if both forward and backward scattering are important, two Dirac delta functions are needed. In problems without collimated radiation, those phase functions can be easily handled using the discrete ordinates method (DOM). However, when collimated irradiation is present, the DOM cannot be applied using the decomposition of the radiation intensity into a diffuse and a collimated component. A new formulation of the DOM to solve radiative transfer problems with collimated irradiation in anisotropically scattering media with such approximate phase functions is described in this work. The proposed method is based on a decomposition of the radiation intensity into three components, namely a collimated, a backscattered collimated and a diffuse component. The method is applied to problems without and with collimated radiation and its accuracy is assessed through comparison with results obtained from Monte Carlo simulations. When a linear approximate phase function is used, the new formulation of the DOM yields accurate results without collimated irradiation, but does not perform so well with collimated irradiation, particularly for media with an optical thickness of the order of unity or lower. However, if the linear approximation is replaced by a quadratic one, the results are significantly improved.

Second-Order Synchroextracting Transform with Application to Fault Diagnosis

Synchrosqueezing transform (SST) is a currently proposed novel postprocessing time-frequency (TF) analysis tool. It has been widely shown that SST is able to improve TF representation. However, so far, how to improve the TF resolution while ensuring the accuracy of signal reconstruction is still an open question, particularly for the vibration signal with time-varying instantaneous frequency (IF) characteristics, due to the fact that the vibration signals of mechanical equipment usually contain many types of noise generated by harsh operating conditions, and the SST will mix these noise into the real signal. Our first contribution is using the Gaussian modulated linear chirp (GMLC) signal model to represent the general nonstationary signals. The GMLC signal model can more accurately represent the time-varying nonstationary signal, compared with the SST signal model composed of linear phase function and constant amplitude. Our second contribution in this work is proposing a method to improve the TF resolution and reconstruction accuracy for nonstationary signals with time-varying IF, which we coined the second-order synchroextracting transform (SET2). In SET2, we apply the GMLC to deduce the nonstationary signal model and then only use the energy at the IF to characterize the TF distribution, which improves the TF while reducing the impact of noise on the real signal.

Sensitivity study on the contribution of scattering by randomly oriented nonspherical hydrosols to linear polarization in clear to semi-turbid shallow waters.

The influence of hydrosol nonsphericity on the polarization characteristics of light under water is investigated by combining accurate single-scattering models for randomly oriented spheroidal scatterers with a radiative transfer model that employs Stokes formalism and considers refraction of direct unpolarized solar radiation and 100% linearly polarized radiation at the air-water interface followed by single scattering. Variations in what we call the "linear polarization phase function" (the degree of linear polarization as a function of scattering angle and the angle of linear polarization as a function of scattering angle) are examined for a wide range of spheroid aspect ratios and complex refractive indices of hydrosols. Implications for polarization-sensitive marine organisms and for remote sensing of the marine environment are discussed.

Sensitivity study on the effect of the optical and physical properties of coated spherical particles on linear polarization in clear to semi-turbid waters

The influence of internal inhomogeneities within hydrosol particles on the polarization characteristics of light is investigated by combining an accurate coated sphere (core-shell) single-scattering model with a radiative transfer model that employs Stokes formalism and considers refraction of direct solar radiation at the air-water interface followed by single scattering. A Junge particle size distribution is assumed. Variations in what we call the "linear polarization phase function" (the degree of linear polarization as a function of scattering angle and the E-vector orientation as a function of scattering angle) are examined as a function of variations in the characteristics of the hydrosol particles. An extensive sensitivity study on the influence of variations in the real and imaginary parts of the refractive index of both the core and shell of the hydrosol particles and on the influences of variations in the ratio between the core radius and shell radius is conducted, varying the values of these parameters over the entire parameter space documented in the literature for actual hydrosol particles. In addition, calculations are conducted for specific parameter combinations in order to demonstrate the influence of some of the most important groups of hydrosols, namely, phytoplankton, gas bubbles, carbonaceous hydrosols, and mineral hydrosols, on the polarization field under water. Variations as a function of solar zenith angle are also investigated. Due to the assumption of single scattering, the results presented are relevant to conditions of low wind speed and a low scattering optical depth and/or low single-scattering albedo within the water body (clear to semi-turbid waters at shallow geometric depths and/or moderate to strong absorption within the water body) outside of Snell's window. Possible implications for aquatic animal polarization vision, for light polarization pollution, and for remote sensing are discussed.

Estimation and compensation methods of time delay and phase offset in hybrid filter bank DACs

This Letter studies the non-ideal analogue filters performance, time delay and phase offset in hybrid filter bank (HFB) DACs. After compensating the non-ideal filters performance using a digital signal, a linear phase function containing time delay and phase was deduced based on the auto-power spectrum of the HFB DAC output in the overlapped frequency band. ‘Three-point’ method was used to simultaneously estimate these two errors by unwrapping the phase affected the principal value interval of the inverse trigonometric function. These two errors were compensated by the digital pre-distortion technique. Finally, simulation results were provided to verify the effectiveness of this proposed method.

Radiative characteristics of a particle using a bump and pit sphere model

Using the particle superposition method, we constructed a bump and pit sphere model that randomly distributes bumps and pits onto the surface of a wavelength-sized host sphere to simulate a rough particle. We studied both the influence of the imaginary part of the refractive index and the number of dusted grains on the particle's radiative properties and found that the linear polarization and phase function of the particle were sensitive to these two parameters. Changes in the imaginary part of the refractive index caused a regular change in the linear polarization, which was reflected in the number of positive peaks and negative troughs. In addition, the number of dusted grains on the surface of the particle changed the absolute value of the linear polarization, because the linear polarization of a particle with many grains on its surface has a smaller absolute value than a particle with few grains. The backward-scattering region of the phase function became significantly smoother for non-dielectric particles than for dielectric particles. Finally, we analyzed the radiative characteristics of polydisperse compound particles. We found that the optical effects of the ratio of pit number to bump number on the polydisperse compound particles are rather weak, if not negligible.

Rotation of cometary nuclei: new light curves and an update of the ensemble properties of Jupiter-family comets

We report new lightcurves and phase functions for nine Jupiter-family comets (JFCs). They were observed in the period 2004-2015 with various ground telescopes as part of the Survey of Ensemble Physical Properties of Cometary Nuclei (SEPPCoN) as well as during devoted observing campaigns. We add to this a review of the properties of 35 JFCs with previously published rotation properties. The photometric time-series were obtained in Bessel R, Harris R and SDSS r' filters and were absolutely calibrated using stars from the Pan-STARRS survey. This specially-developed method allowed us to combine data sets taken at different epochs and instruments with absolute-calibration uncertainty down to 0.02 mag. We used the resulting time series to improve the rotation periods for comets 14P/Wolf, 47P/Ashbrook-Jackson, 94P/Russell, and 110P/Hartley 3 and to determine the rotation rates of comets 93P/Lovas and 162P/Siding-Spring for the first time. In addition to this, we determined the phase functions for seven of the examined comets and derived geometric albedos for eight of them. We confirm the known cut-off in bulk densities at $\sim$0.6 g $\mathrm{cm^{-3}}$ if JFCs are strengthless. Using the model of Davidsson (2001) for prolate ellipsoids with typical density and elongations, we conclude that none of the known JFCs require tensile strength larger than 10-25 Pa to remain stable against rotational instabilities. We find evidence for an increasing linear phase function coefficient with increasing geometric albedo. The median linear phase function coefficient for JFCs is 0.046 mag/deg and the median geometric albedo is 4.2 per cent.

L2-estimates for singular oscillatory integral operators

In this note we study singular oscillatory integrals with linear phase function over hypersurfaces which may oscillate, and prove estimates of L2↦L2 type for the operator, as well as for the corresponding maximal function. If the hypersurface is flat, we consider a particular class of a nonlinear phase functions, and apply our analysis to the eigenvalue problem associated with the Helmholtz equation in R3.

Phase and amplitude tracking for seismic event separation

We have developed a method to decompose seismic records into atomic events, each defined by a smooth phase function and a smooth amplitude function. This decomposition is intrinsically nonlinear and calls for a nonconvex least-squares optimization formulation, along the lines of full-waveform inversion. To overcome the lack of convexity, we have developed an iterative refinement-expansion scheme to initialize and track the phase and amplitude for each atomic event. For short, we called the method phase tracking. The initialization is carried out by applying multiple signal classification to a few seed traces in which events can be separated and identified by their arrival times and amplitudes. We then construct the initial solution at the seed traces using linear phase functions from the arrival times and constant amplitude functions, assuming the medium is mostly dispersion free. We refine this initial solution to account for dispersion and imperfect knowledge of the wavelet at the seed traces by fitting the observed data using a gradient descent method. The resulting phase and amplitude functions are then carefully expanded across the traces in an adequately smooth way to match the whole data record. We have evaluated the proposed method on two synthetic records and a field record. Because the parametrization of the seismic events is physically meaningful, it also enables a simple form of bandwidth extension of the observed shot record to unobserved low and high frequencies. We tested this procedure on the same shot records. Bandwidth extension is in principle helpful to initialize full-waveform inversion with frequency sweeps and enhanced its resolution.

Time-domain solution to the radiative transfer equation in an infinite turbid medium with linearly anisotropic scattering.

A time-dependent solution of the transport equation with an instantaneous point source is considered by the multiple collision method developed by Kholin [Zh. Vych. Mat. i Mat. Fys.4, 1126 (1964)]. For a linear phase function, Kholin's solution is found to be considerably simplified and can be expressed through a single integral. The shape of the time-dependent angle-averaged scattering intensity is shown to depend on the distance from the source. For short distances, the angle-averaged scattering intensity decays monotonically with time. For longer distances, the angle-averaged intensity exhibits two peaks. The first infinite peak occurs at the wavefront and is followed by a narrow decaying tail. The second peak occurs at approximately the position of the diffusion peak but is higher than the diffusion peak. At large distances, the shape of the intensity after the wavefront arrival is well approximated by the diffusion shape. Good agreement with Monte Carlo simulation is demonstrated.

HUBBLE SPACE TELESCOPE OBSERVATIONS OF THE NUCLEUS OF COMET C/2012 S1 (ISON)

We report on the analysis of several sequences of broadband visible images of comet C/2012 S1 (ISON) taken with the Wide Field Camera 3 of the Hubble Space Telescope on 2013 April 10, May 8, October 9, and November 1 in an attempt to detect and characterize its nucleus. Whereas the overwhelming coma precluded the detection of the nucleus in the first two sequences, the contrast was sufficient in early October to unambiguously retrieve the signal from the nucleus. Two images taken within a few minutes led to similar V magnitudes for the nucleus of 21.97 and 22.0 with a 1σ uncertainty of 0.065. Assuming a standard value for the geometric albedo (0.04) and a linear phase function with a coefficient of 0.04 mag deg{sup –1}, these V values imply that the nucleus radius is 0.68 ± 0.02 km. Although this result does depend on these two assumptions, we argue that the radius most likely lies in the range 0.6-0.9 km. This result is consistent with the constraints derived from the water production rates reported by Combi et al. The last sequence of images in 2013 November revealed temporal variation of the innermost coma. If attributed to a single rotating jet, thismore » coma brightness variation suggests the rotational period of the nucleus may be close to ∼10.4 hr.« less

Single-scattering solutions to radiative transfer in infinite turbid media

An instantaneous point source generating a light field in an infinite turbid medium with anisotropic individual scatterers is considered. Single-scattering solutions are sought as the first term of a series expansion of the solution of the radiative transfer equation in successive scattering orders. A simple formula for a single-scattering solution in media with an arbitrary axially symmetric phase function is derived. Applications of this formula are shown for the Henyey-Greenstein phase function, an ellipsoidal phase function, and a linear phase function. In addition, the single-scattering term of the RTE solution derived by Kholin [Zh. Vych. Mat. i Mat. Fys.4, 1126 (1964)] for media with a phase function represented by a finite series in Legendre polynomials is considered in more detail.

Sensitivity study on the effects of hydrosol size and composition on linear polarization in absorbing and nonabsorbing clear and semi-turbid waters

A full Mie scattering subroutine is employed to calculate what we call the linear polarization phase function (LPPF; percent polarization and e-vector orientation of radiation as a function of scattering angle) that results from refraction of the direct solar beam from air into water followed by single scattering by spherical hydrosols. The separate effects of refraction at the air-water interface, hydrosol size, the real and imaginary parts of the hydrosol refractive index, and absorption by the surrounding medium (water) on the LPPF are investigated. All of the above factors are found to alter the LPPF, changing the value of the maximum percent polarization (P(max)), the location of P(max), the number of fluctuations in the LPPF, or the location of the neutral points (points of 0 percent polarization), though absorption by the surrounding medium is found to have only a minimal effect. The character and extent of the influence on the LPPF is found to depend on the scattering regime (Rayleigh, Mie, or geometric optics). We conclude that in calculating underwater polarization, it is important to take into consideration Mie scattering even in relatively clear waters. We also find a coupling between the partial polarization and the e-vector orientation, which suggests that for some polarization-based visual tasks, only one of these would suffice. Other implications for aquatic animal polarization vision are discussed.

Entire-Domain Basis Functions for Scattering From Large Curved Surfaces Formulated by Transformation Optics

A complete set of entire-domain basis functions are introduced for the analysis of scattering from bodies with curved surfaces; they are defined via the application of a generalization of the Shannon sampling theorem. These basis functions are defined for curved patches with arbitrary contours, via a three-step procedure. First, the curved patch is mapped, via Transformation Optics, onto a flat parametric domain surrounded by a virtual anisotropic inhomogeneous space. Next, linear-phase functions are defined on the parametric flat domain; the sufficient and non-redundant number of functions is found by using a spectral domain “completeness relationship” of the delta function. Finally, the back-transformation from flat anisotropic to curved isotropic space yields the basis functions for the curved patch. The number of basis functions so obtained matches the degrees of freedom of the field known in the literature. Numerical results show the effectiveness of the representation for both fields and currents.

PHYSICAL PROPERTIES OF MAIN-BELT COMET 176P/LINEAR

We present a physical characterization of comet 176P/LINEAR, the third discovered member of the new class of main-belt comets, which exhibit cometary activity but are dynamically indistinguishable from main-belt asteroids. Observations show the object exhibiting a fan-shaped tail for at least one month in late 2005, but then becoming inactive in early 2006. During this active period, we measure broadband colors of B − V = 0.63 ± 0.02, V − R = 0.35 ± 0.02, and R − I = 0.31 ± 0.04. Using data from when the object was observed to be inactive, we derive best-fit IAU phase function parameters of H = 15.10 ± 0.05 mag and G = 0.15 ± 0.10, and best-fit linear phase function parameters of m(1, 1, 0) = 15.35 ± 0.05 mag and β = 0.038 ± 0.005 mag deg−1. From this baseline phase function, we find that 176P exhibits a mean photometric excess of ∼30% during its active period, implying an approximate total coma dust mass of Md ∼ (7.2 ± 3.6) × 104 kg. From inactive data obtained in early 2007, we find a rotation period of Prot = 22.23 ± 0.01 hr and a peak-to-trough photometric range of Δm ∼ 0.7 mag. Phasing our photometric data from 176P's 2005 active period to this rotation period, we find that the nucleus exhibits a significantly smaller photometric range than in 2007 that cannot be accounted for by coma damping effects, and as such, are attributed by us to viewing geometry effects. A detailed analysis of these geometric effects showed that 176P is likely to be a highly elongated object with an axis ratio of 1.8 < b/a < 2.1, an orbital obliquity of ε ∼ 60°, and a solstice position at a true anomaly of νo = 20° ± 20°. Numerical modeling of 176P's dust emission found that its activity can only be reproduced by asymmetric dust emission, such as a cometary jet. We find plausible fits to our observations using models assuming ∼10 μm dust particles continuously emitted over the period during which 176P was observed to be active, and a jet direction of 180° ≲ αjet ≲ 120° and δjet ≈ −60°. We do not find good fits to our observations using models of impulsive dust emission, i.e., what would be expected if 176P's activity was an ejecta cloud resulting from an impact into non-volatile asteroid regolith. Since for a rotating body, the time-averaged direction of a non-equatorial jet is equivalent to the direction of the nearest rotation pole, we find an equivalent orbital obliquity of 50° ≲ ε ≲ 75°, consistent with the results of our light curve analysis. Furthermore, the results of both our light curve analysis and dust modeling analysis are consistent with the seasonal heating hypothesis used to explain the modulation of 176P's activity. Additional observations are highly encouraged to further characterize 176P's active behavior as the object approaches perihelion on 2011 July 1.