Direction Cosine Space Research Articles

Theoretical derivation and application of empirical Harvey scatter model.

Starting from the Rayleigh-Rice perturbation theory, this paper derives the empirical Harvey scatter model and ABg scatter model applied extensively in optical analysis software packages and verifies the shift-invariant behavior of the scattered radiance in direction cosine space. Using data obtained from multi-wavelength laser scatterometer on carbon nanotube black coating and pineblack coating, we establish the polynomial model based on the sine of the scattering angle plus the sine of the specular reflection angle, i.e., sin θs+sin θ0 and the dual-Harvey model based on sin θs-sin θ0 , respectively. The models are in good accordance with the experimental data and further extend the valid range of empirical models.

Optical diffraction properties of three superimposed self-organized nanostructures induced by a laser process.

Controlling the diffraction properties of materials over a large area holds great promise for a wide range of optical applications. Laser-based techniques have emerged as a viable solution to address this need. Here, we present the diffraction properties of laser-induced self-organized structures, which consist of three interlaced grating-like structures: self-organized nanoparticles, self-organized cracks, and laser marking lines. Under normal incidence external illumination, the sample exhibits an asymmetric diffraction pattern. However, when the incidence angle is tilted, circular diffraction patterns are observed in the plane perpendicular to both the sample and the incidence plane. These phenomena are attributed to the combination effect of the diffraction gratings. To elucidate the underlying physics of multiple diffraction, we use rigorous coupled-wave analysis (RCWA) and grating equations written in direction cosine space, extended to account for the presence of three superimposed gratings. Exploiting the laser-induced diffraction properties of these samples may have great potential for various industrial implementations, including security, display, and design.

Optimization of gratings in a diffractive waveguide using relative-direction-cosine diagrams.

Designing diffractive waveguides for head-mounted displays requires wide-angle conical diffraction analysis of multiple gratings. In this work, diffractive waveguide design using the relative direction cosine space, which extends the direction cosine space to a relative space involving refractive indices and can describe grating diffraction through various media, is demonstrated. A diffractive waveguide was fabricated with grating periods of 382 and 270 nm, which generated a monochromatic virtual image image in green light (520 nm). The maximum field of view was measured as 39° with 0.5° deviation from the center of view.

Design method toward creating a three-dimensional asymmetric compound parabolic concentrator

Ideal three-dimensional concentration has been sought for decades in the solar and optical worlds. Nonimaging optics has been able to achieve this thermodynamic maximum concentration limit with certain symmetrical designs, but three-dimensional concentration in which acceptance angles are not symmetric has been a challenge for researchers. Our study outlines a new design approach that hopes to move toward the development of an ideal 3D asymmetric concentrator based on the current asymmetric compound parabolic concentrator (ACPC) design principles. A geometric scheme to revolve a variable ACPC curve about a central axis is presented. The 3D ACPC is compared with a standard compound parabolic concentrator (CPC) in a tilted and angular truncated case. Each design was characterized for performance by examining the angular acceptance/illumination region in angular direction cosine space. Ideal performance was demonstrated for the 3D ACPC design for an off axis circular Lambertian acceptance region in direction cosine plots, whereas the CPC designs failed to maintain a circular Lambertian distribution. High-aperture tilt angles demonstrated an interesting “flaring” of the aperture, which caused a decline in ideal performance. Despite design deformations at high-aperture angles, the 3D ACPC design method presented here is a step toward ideal 3D asymmetric concentration.

Comparison of microfacet BRDF model to modified Beckmann-Kirchhoff BRDF model for rough and smooth surfaces.

A popular class of BRDF models is the microfacet models, where geometric optics is assumed. In contrast, more complex physical optics models may more accurately predict the BRDF, but the calculation is more resource intensive. These seemingly disparate approaches are compared in detail for the rough and smooth surface approximations of the modified Beckmann-Kirchhoff BRDF model, assuming Gaussian surface statistics. An approximation relating standard Fresnel reflection with the semi-rough surface polarization term, Q, is presented for unpolarized light. For rough surfaces, the angular dependence of direction cosine space is shown to be identical to the angular dependence in the microfacet distribution function. For polished surfaces, the same comparison shows a breakdown in the microfacet models. Similarities and differences between microfacet BRDF models and the modified Beckmann-Kirchhoff model are identified. The rationale for the original Beckmann-Kirchhoff F(bk)(2) geometric term relative to both microfacet models and generalized Harvey-Shack model is presented. A modification to the geometric F(bk)(2) term in original Beckmann-Kirchhoff BRDF theory is proposed.

Experimental analysis of bidirectional reflectance distribution function cross section conversion term in direction cosine space

Of the many classes of bidirectional reflectance distribution function (BRDF) models, two popular classes of models are the microfacet model and the linear systems diffraction model. The microfacet model has the benefit of speed and simplicity, as it uses geometric optics approximations, while linear systems theory uses a diffraction approach to compute the BRDF, at the expense of greater computational complexity. In this Letter, nongrazing BRDF measurements of rough and polished surface-reflecting materials at multiple incident angles are scaled by the microfacet cross section conversion term, but in the linear systems direction cosine space, resulting in great alignment of BRDF data at various incident angles in this space. This results in a predictive BRDF model for surface-reflecting materials at nongrazing angles, while avoiding some of the computational complexities in the linear systems diffraction model.

Preparing the generalized Harvey-Shack rough surface scattering method for use with the discrete ordinates method.

The paper shows how to implement the generalized Harvey-Shack (GHS) method for isotropic rough surfaces discretized in a polar coordinate system and approximated using Fourier series. This is particularly relevant for the use of the GHS method as a boundary condition for radiative transfer problems in slab geometries, where the discrete ordinates method can be applied to solve the problem. Furthermore, such an implementation is a more convenient discretization of the problem than the traditional direction cosine space that has its strengths in analytical problems and intuitive understanding (mainly due to its translation invariance). A computer implementation of scattering from a Gaussian rough surface with Gaussian autocovariance written in Python is included at the end of the paper.

Change Detection from Landsat TM Images Using Change Vector Analysis

Change detection is a process of extracting, analyzing, and defining change information from remote sensing imageries. At present, remote sensing change detection methods are mainly classified into two categories, one based on the spectral characteristics of the type of approach, and the other is spectral change vector analysis. In this paper, a simplified threshold variable step-size search which can be used on determining changes in the threshold vector is adopted, as well as the supervised classification technique in the direction cosine space with the type of focal point in the initial assay vector remote sensing images. Results are discussed in the last part of this paper, which show that CVA can extract change information effectively in our study area of Wuhan city.

Change Detection Using Change Vector Analysis from Landsat TM Images in Wuhan

Abstract Spectral Change Vector Analysis (CVA) is based on multi-temporal images. In this paper, a dichotomy search which can be used on detecting changes in the threshold vector is adopted. Meanwhile, a supervised classification technique is used in the direction cosine space with the type of central point in the initial assay vector remote sensing images. Results are discussed in the last part of this paper, which show that CVA can extract change information effectively in our study area of Wuhan city.

Pushing concentration of stationary solar concentrators to the limit

We give the theoretical limit of concentration allowed by nonimaging optics for stationary solar concentrators after reviewing sun- earth geometry in direction cosine space. We then discuss the design principles that we follow to approach the maximum concentration along with examples including a hollow CPC trough, a dielectric CPC trough, and a 3D dielectric stationary solar concentrator which concentrates sun light four times (4x), eight hours per day year around.

Nonparaxial scalar treatment of sinusoidal phase gratings

Scalar diffraction theory is frequently considered inadequate for predicting diffraction efficiencies for grating applications where lambda/d>0.1. It has also been stated that scalar theory imposes energy upon the evanescent diffracted orders. These notions, as well as several other common misconceptions, are driven more by an unnecessary paraxial approximation in the traditional Fourier treatment of scalar diffraction theory than by the scalar limitation. By scaling the spatial variables by the wavelength, we have previously shown that diffracted radiance is shift invariant in direction cosine space. Thus simple Fourier techniques can now be used to predict a variety of wide-angle (nonparaxial) diffraction grating effects. These include (1) the redistribution of energy from the evanescent orders to the propagating ones, (2) the angular broadening (and apparent shifting) of wide-angle diffracted orders, and (3) nonparaxial diffraction efficiencies predicted with an accuracy usually thought to require rigorous electromagnetic theory.

Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory

Most authors include a paraxial (small-angle) limitation in their discussion of diffracted wave fields. This paraxial limitation severely limits the conditions under which diffraction behavior is adequately described. A linear systems approach to modeling nonparaxial scalar diffraction theory is developed by normalization of the spatial variables by the wavelength of light and by recognition that the reciprocal variables in Fourier transform space are the direction cosines of the propagation vectors of the resulting angular spectrum of plane waves. It is then shown that wide-angle scalar diffraction phenomena are shift invariant with respect to changes in the incident angle only in direction cosine space. Furthermore, it is the diffracted radiance (not the intensity or the irradiance) that is shift invariant in direction cosine space. This realization greatly extends the range of parameters over which simple Fourier techniques can be used to make accurate calculations concerning wide-angle diffraction phenomena. Diffraction-grating behavior and surface-scattering effects are two diffraction phenomena that are not limited to the paraxial region and benefit greatly from this new development.

Description of Diffraction Grating Behavior in Direction Cosine Space

It is well known that the angular separation of non-paraxial diffracted orders from a linear grating varies drastically with incident angle. Furthermore, for oblique incident angles (conical diffraction), it is rather cumbersome both analytically and graphically to describe the number and angular position of the various propagating orders. One can readily demonstrate that wide-angle diffraction phenomena (including conical diffraction from gratings) are shift-invariant with respect to incident angle in direction cosine space. Only when the grating equation is expressed in terms of the direction cosines of the propagation vectors of the incident beam and the diffracted orders can we apply the Fourier techniques resulting from linear systems theory. This formulation has proven extremely useful for small-angle diffraction phenomena and in modern, image formation theory. New insight and an intuitive understanding of diffraction grating behavior results from a simple direction cosine diagram.