Arbitrary Phase Function Research Articles

Spin-decoupling of vertical cavity surface-emitting lasers with complete phase modulation using on-chip integrated Jones matrix metasurfaces

Polarization response of artificially structured nano-antennas can be exploited to design innovative optical components, also dubbed “vectorial metasurfaces”, for the modulation of phase, amplitude, and polarization with subwavelength spatial resolution. Recent efforts in conceiving Jones matrix formalism led to the advancement of vectorial metasurfaces to independently manipulate any arbitrary phase function of orthogonal polarization states. Here, we are taking advantages of this formalism to design and experimentally validate the performance of CMOS compatible Jones matrix metasurfaces monolithically integrated with standard VCSELs for on-chip spin-decoupling and phase shaping. Our approach enables accessing the optical spin states of VCSELs in an ultra-compact way with previously unattainable phase controllability. By exploiting spin states as a new degree of freedom for laser wavefront engineering, our platform is capable of operating and reading-out the spin-momentum of lasers associated with injected spin carriers, which would potentially play a pivotal role for the development of emerging spin-optoelectronic devices.

Once‐more scattered next event estimation for volume rendering

AbstractWe present a Monte Carlo path tracing technique to sample extended next event estimation contributions in participating media: we consider one additional scattering vertex on the way to the next event, accounting for focused blur, resulting in visually interesting image features. Our technique is tailored to thin homogeneous media with strongly forward scattering phase functions, such as water or atmospheric haze. Previous methods put emphasis on sampling transmittances or geometric factors, and are either limited to isotropic scattering, or used tabulation or polynomial approximation to account for some specific phase functions. We will show how to jointly importance sample the product of an arbitrary phase function with analytic sampling in the solid angle domain and the two reciprocal squared distance terms of the adjacent edges of the transport path. The technique is fast and simple to implement in an existing rendering system. Our estimator is designed specifically for forward scattering, so the new technique has to be combined with other estimators to cover the backward scattering contributions.

Decoupling Frequencies, Amplitudes and Phases in Nonlinear Optics

In linear optics, light fields do not mutually interact in a medium. However, they do mix when their field strength becomes comparable to electron binding energies in the so-called nonlinear optical regime. Such high fields are typically achieved with ultra-short laser pulses containing very broad frequency spectra where their amplitudes and phases are mutually coupled in a convolution process. Here, we describe a regime of nonlinear interactions without mixing of different frequencies. We demonstrate both in theory and experiment how frequency domain nonlinear optics overcomes the shortcomings arising from the convolution in conventional time domain interactions. We generate light fields with previously inaccessible properties by avoiding these uncontrolled couplings. Consequently, arbitrary phase functions are transferred linearly to other frequencies while preserving the general shape of the input spectrum. As a powerful application, we introduce deep UV phase control at 207 nm by using a conventional NIR pulse shaper.

Application of general invariance relations reduction method to solution of radiation transfer problems

A brief analysis of different properties and principles of invariance to solve a number of classical problems of the radiation transport theory is presented. The main ideas, constructions, and assertions used in the general invariance relations reduction method are described in outline. The most important distinctive features of this general method of solving a wide enough range of problems of the radiation transport theory and mathematical physics are listed. To illustrate the potential of this method, a number of problems of the scalar radiative transfer theory have been solved rigorously in the article. The main stages of rigorous derivations of asymptotical formulas for the smallest in modulo elements of the discrete spectrum and the eigenfunctions, corresponding to them, of the characteristic equation for the case of an arbitrary phase function and almost conservative scattering are described. Formulas of the same type for the azimuthal averaged reflection function, the plane and spherical albedos have been obtained rigorously. New analytical representations for the reflection function, the plane and spherical albedos have been obtained, and effective algorithms for calculating these values have been offered for the case of a practically arbitrary phase function satisfying the Hölder condition. New analytical representation of the «surface» Green function of the scalar radiative transfer equation for a semi-infinite plane-parallel conservatively scattering medium has been found. The deep regime asymptotics of the “volume” Green function has been obtained for the case of a turbid medium of cylindrical form.

Composite Method of Forming Approximating Distributions with an Arbitrary Phase Function

In this article we consider an approach to representation of distributions of probabili-ties in the form of the two-level composition of an integral kernel and a phase function which is generalization of the concept of density of random parameter distribution. Possibilities of giper-delta approximation of the phase function and its interrelation with the formation of phase-type distributions are shown. The method of approximating distributions formation on the basis of the arbitrary phase function by the method of derivatives is offered.

Solving the radiative transfer equation with a mathematical particle method.

We solve the radiative transfer equation (RTE) using a recently proposed mathematical particle method, originally developed for solving general functional equations. We show that, in the case of the RTE, it gives several advantages, such as handling arbitrary boundary conditions and phase functions and avoiding numerical instability in strongly forward-scattering media. We also solve the RTE, including fluorescence, and an example is shown with a fluorescence cascade where light is absorbed and emitted in several steps. We show that the evaluated particle method is straightforward to implement, which is in contrast with many traditional RTE solvers, but a potential drawback is the tuning of the method parameters.

Photometry of dark atmosphereless planetary bodies: an efficient numerical model

We present a scattering model for regolith-covered Solar System bodies. It can be used to compute the intensity of light scattered by a surface consisting of packed, mutually shadowing particles. Our intention is to provide a model in which other researchers can apply in studies of Solar System photometry.Our model is a Lommel–Seeliger type model, representing a medium composed of individual scatterers with small single-scattering albedo. This means that it is suitable for dark regolith surfaces such as the Moon and many classes of asteroids. Our model adds an additional term which takes into account the mutual shadowing between the scatterers. The scatterers can have an arbitrary phase function. We use a numerical ray-tracing simulation to compute the shadowing contribution.We present the model in a form which makes implementing it in existing software straightforward and fast. The model in practice is implemented as files containing pre-computed values of the surface reflection coefficient, which can be loaded into a user's program and used to compute the scattering in the desired viewing geometries. As the usage requires only a little simple arithmetic and a table look-up, it is as fast to use as common analytical models.

Transport of intensity equation: Validity limits of the usually accepted solution

The transport of intensity equation (TIE) is the basis of a powerful non-interferometric technique for phase retrieval. When intensity gradients are present, the generalized belief is that the solution of the TIE (i.e. the phase distribution) can be found by solving two Poisson equations. In the present paper we will demonstrate that this expression for the TIE solution holds when intensity and phase gradients are parallel (or null), but in general the usually accepted solution is not correct. We perform simulations with arbitrary intensity and phase functions in order to show the phase errors derived from the use of this expression.

The instantaneous fluctuation theorem

We give a derivation of a new instantaneous fluctuation relation for an arbitrary phase function which is odd under time reversal. The form of this new relation is not obvious, and involves observing the system along its transient phase space trajectory both before and after the point in time at which the fluctuations are being compared. We demonstrate this relation computationally for a number of phase functions in a shear flow system and show that this non-locality in time is an essential component of the instantaneous fluctuation theorem.

Historygrams: Enabling Interactive Global Illumination in Direct Volume Rendering using Photon Mapping.

In this paper, we enable interactive volumetric global illumination by extending photon mapping techniques to handle interactive transfer function (TF) and material editing in the context of volume rendering. We propose novel algorithms and data structures for finding and evaluating parts of a scene affected by these parameter changes, and thus support efficient updates of the photon map. In direct volume rendering (DVR) the ability to explore volume data using parameter changes, such as editable TFs, is of key importance. Advanced global illumination techniques are in most cases computationally too expensive, as they prevent the desired interactivity. Our technique decreases the amount of computation caused by parameter changes, by introducing Historygrams which allow us to efficiently reuse previously computed photon media interactions. Along the viewing rays, we utilize properties of the light transport equations to subdivide a view-ray into segments and independently update them when invalid. Unlike segments of a view-ray, photon scattering events within the volumetric medium needs to be sequentially updated. Using our Historygram approach, we can identify the first invalid photon interaction caused by a property change, and thus reuse all valid photon interactions. Combining these two novel concepts, supports interactive editing of parameters when using volumetric photon mapping in the context of DVR. As a consequence, we can handle arbitrarily shaped and positioned light sources, arbitrary phase functions, bidirectional reflectance distribution functions and multiple scattering which has previously not been possible in interactive DVR.

Photon diffusion near the point-of-entry in anisotropically scattering turbid media

From astronomy to cell biology, the manner in which light propagates in turbid media has been of central importance for many decades. However, light propagation near the point-of-entry in turbid media has never been analytically described, until now. Here we report a straightforward and accurate method that overcomes this longstanding, unsolved problem in radiative transport. Our theory properly treats anisotropic photon scattering events and takes the specific form of the phase function into account. As a result, our method correctly predicts the spatially dependent diffuse reflectance of light near the point-of-entry for any arbitrary phase function. We demonstrate that the theory is in excellent agreement with both experimental results and Monte Carlo simulations for several commonly used phase functions.

Instrumentation to measure the backscattering coefficient b_b for arbitrary phase functions

A single detector instrument concept that collects scattered light over the full range of backscattering angles is described. Its light collection aperture is designed so as to introduce a sin θ factor into the collection probability. Hence, the instrument is exactly a b(b) meter; it directly measures b(b), not a proxy for it. For an infinitesimal aperture to the detector, the instrument would give b(b) exactly; for a finite aperture (e.g., 1.26 cm(2)), it would typically give b(b) to an accuracy of a few tenths of 1%. The instrumentation itself is as simple as that of the well-known fixed-angle meters-it projects a beam of light into the medium and collects backscattered light with a single detector; the differences are the position of the detector and the shape/orientation of the entrance aperture to the detector.

Differential Approximations to the Radiation Transfer Equation by Chapman–Enskog Expansion

Direct numerical solution of the radiation transfer equation is often easier than implementation of its differential approximations with their cumbersome boundary conditions. Nevertheless, these approximations are still used, for example, in theoretical analysis. The existing approach to obtain a differential approximation based on expansion in series of the spherical harmonics is revised and expansion in series of the eigenfunctions of the scattering integral is proposed. A system of eigenfunctions is obtained for an arbitrary phase function, and explicit differential approximations are built up to the third Chapman–Enskog order. The results are tested by its application to the problem of a layer. The third-order Chapman–Enskog approximation is found to match the boundary conditions better than the first-order one and gives considerably more accurate value for the heat flow. The accuracy of the both first- and third-order heat flows generally increases with the optical thickness. In addition, the third-order heat flow tends to the rigorous limit value when the optical thickness tends to zero.

Physical interpretation and accuracy of the Kubelka–Munk theory

Kubelka–Munk theory is used extensively in a number of industrial applications including colour matching. The main aim of this paper is the derivation of alternative analytical equations, which are more accurate than correspondent Kubelka–Munk formulae for light scattering materials of a finite optical depth. The derived equations are based on asymptotic radiative transfer theory valid for optically thick turbid media with arbitrary phase functions and single scattering albedos.

Radiative transport equation in rotated reference frames

A novel method for solving the linear radiative transport equation (RTE) in a three-dimensional homogeneous medium is proposed and illustrated with numerical examples. The method can be used with an arbitrary phase function with the constraint that it depends only on the angle between the angular variables and . This assumption corresponds to spherically symmetric (on average) random medium constituents. Boundary conditions are considered in the slab and half-space geometries. The approach developed in this paper is spectral. It allows for the expansion of the solution to the RTE in terms of analytical functions of angular and spatial variables to relatively high orders. The coefficients of this expansion must be computed numerically. However, the computational complexity of this task is much smaller than in the standard method of spherical harmonics. The solutions obtained are especially convenient for solving inverse problems associated with radiative transfer.

Generalized Eddington analytical model for azimuthally dependent radiance simulation in stratified media

A fast analytical radiative transfer model to account for propagation of unpolarized monochromatic radiation in random media with a plane-parallel geometry is presented. The model employs an Eddington-like approach combined with the delta phase-function transformation technique. The Eddington approximation is extended in a form that allows us to unfold the azimuthal dependence of the radiance field. A first-order scattering correction to the azimuth-dependent Eddington radiative model solution is also performed to improve the model accuracy for low-scattering media and flexibility with respect to use of explicit arbitrary phase functions. The first-order scattering-corrected solution, called the generalized Eddington radiative model (GERM), is systematically tested against a numerical multistream discrete ordinate model for backscattered radiance at the top of the medium. The typical mean accuracy of the GERM solution is generally better than 10% with a standard deviation of 20% for radiance calculations over a wide range of independent input optical parameters and observation angles. GERM errors are shown to be comparable with the errors due to an input parameter uncertainty of precise numerical models. The proposed model can be applied in a quite arbitrary random medium, and the results are appealing in all cases where speed, accuracy, and/or closed-form solutions are requested. Its potentials, limitations, and further extensions are discussed.

High bandwidth spectral gratings programmed with linear frequency chirps

Temporally overlapped linear frequency chirps (TOLFCs), a novel method for programming a spatial-spectral grating, are used to demonstrate grating bandwidths in excess of 2 GHz. The proper operating conditions for this method are discussed along with grating efficiency measurements. A Maxwell–Bloch simulator, which incorporates arbitrary phase functions, is used to simulate this programming method and to investigate the grating dynamics of accumulated programming using TOLFCs.

Multiple Scattering of Light in Regolith-like Media: Geometric Optics Approximation

We present a numerical model that allows us to calculate the contribution of a specified scattering order in the geometric optics approximation for media composed of large particles with an arbitrary phase function. It has been demonstrated that the correlated propagation of the incident and emergent rays, which is disregarded in the classical radiative-transfer theory, markedly affects the contributions of different orders of scattering, especially the first-order scattering. If the theory describing the photometric properties of regolith-like media ignores the shadowing effect, the errors of its application may reach several tens of percent even for bright surfaces. The packing density of a medium essentially influences the phase dependence of the first-order scattering, although its effect on the value and the phase curve of the higher scattering orders is relatively weak. The backscattering peaks calculated on the basis of the Hapke theory are narrower than those obtained from the numerical simulations, because the Hapke theory is an approximate approach.

Non-oscillating solutions to uncoupled Ermakov systems and the semi-classical limit

The amplitude-phase formulation of the Schr\"{o}dinger equation is investigated within the context of uncoupled Ermakov systems, whereby the amplitude function is given by the auxiliary nonlinear equation. The classical limit of the amplitude and phase functions is analyzed by setting up a semiclassical Ermakov system. In this limit, it is shown that classical quantities, such as the classical probability amplitude and the reduced action, are obtained only when the semiclassical amplitude and the accumulated phase are non-oscillating functions respectively of the space and energy variables. Conversely, among the infinitely many arbitrary exact quantum amplitude and phase functions corresponding to a given wavefunction, only the non-oscillating ones yield classical quantities in the limit $\hbar \to 0$.

Continued fraction solution for the radiative transfer equation in three dimensions

Starting from the radiative transfer equation, we obtain an analytical solution for both the free propagator along one of the axes and an arbitrary phase function in the Fourier-Laplace domain. We also find the effective absorption parameter, which turns out to be very different from the one provided by the diffusion approximation. We finally present an analytical approximation procedure and obtain a differential equation that accurately reproduces the transport process. We test our approximations by means of simulations that use the Henyey-Greenstein phase function with very satisfactory results.