STABLE AGGREGATES OF FLEXIBLE ELEMENTS : MAKING A SILK PURSE FROM SOW ’ S EARS
We discuss the application of a ray-based method for wave propagation to problems involving refraction and diffraction. The method assigns a Gaussian contribution to each ray such that their sum is insensitive to the Gaussian’s width. Diffraction is modelled by including contributions from rays scattered by the edges of obstacles. Although the Gaussian components are individually only poor approximations to physical waves, they collectively give accurate solutions even for problems where the wavelength approaches the dimensions of system components. INTRODUCTION A framework for building field estimates based on ray information was introduced in a recent series of papers [1-6]. In this framework, each ray is assigned a field contribution of Gaussian profile. These contributions, however, are not parabasal Gaussian beamlets, as each of them is not required to satisfy the wave equation. They are poor approximations to fields (the “sow’s ears” of the title), but their individual errors are forced to cancel when they are summed, leading to a field estimate of extraordinary accuracy. This estimate is also asymptotically (for small wavelength) insensitive to the width of the individual contributions. The sums are therefore referred to as a stable aggregates of flexible elements (SAFE). Here, we discuss the use of SAFE for modelling refraction and reflection at interfaces, as well as diffraction by opaque obstacles. WAVES IN INHOMOGENEOUS MEDIA Consider a scalar monochromatic field U in a two-dimensional inhomogeneous medium. This field satisfies [k∇ 2 + n(x,z)]U(x,z) = 0, (1) where k = 2π λ is the wave number, and n is the refractive index. We assume that this field propagates essentially towards larger z, so it is sufficient to specify U(x,0) in order to find U at any z > 0. Finding a closed form solution is generally not possible, so numerical schemes are necessary. Some of these schemes, based on rays, are particularly suited for problems where the wavelength is much smaller than any other characteristic length. It is natural to assume that the rays associated to the field described above also travel towards larger z. Each of these rays can then be described by an equation of the form x = X(z) . We assume that a solution to Eq. (1) is associated with a one-parameter family of rays, described collectively by a function X(ξ;z) , where ξ identifies the rays. The propagation of the rays in the inhomogeneous medium is ruled by the equations ∂ X ∂z (ξ;z) = P(ξ;z) H(ξ;z) , ∂P ∂z (ξ;z) = 1 2H(ξ;z) ∂n ∂x [X(ξ;z),z], (2) where P and H are respectively the xand z-components of a vector of length n(X,z) locally tangent to the ray. The optical path length of the rays, measured from a common phase front, is denoted by L(ξ;z). In order to find a ray-based estimate for the field at z > 0, one must first determine from U(x,0) the location and direction of the rays at z = 0, as well as their weight. (This process is discussed in [2].) The rays are then propagated by using Eqs. (2). Finally, the field estimate is constructed from the propagated rays, i.e. from X, P, H, L, and the weight function. For SAFE, this estimate is a sum of ray contributions of the form Uγ (x,z) = w(ξ ,γ;z) g(ξ ,γ; x,z) dξ ∫ , (3) where each contribution has a Gaussian profile in x and is centred at its corresponding ray: g(ξ ,γ; x,z) = k 2π exp − kγ 2 x − X(ξ;z) [ ]2 + ik L(ξ;z) + x − X(ξ;z) [ ] P(ξ;z) { } ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . (4)
Using rays better. I. Theory for smoothly varying media.
We present a method for computing ray-based approximations to optical fields that not only offers unprecedented accuracy but is also accompanied by accessible error estimates. The basic elements of propagation through smooth media, refraction and reflection at interfaces, and diffraction by obstacles give the foundations for the new framework, and the first of these is treated here. The key in each case is that the wave field and any relevant derivatives are expressed consistently as a superposition of delocalized ray contributions. In this way, the mysteries surrounding the sometimes perplexing tenaciousness of ray-based estimates are clearly resolved. Further, an essential degree of freedom in this approach offers an attractive resolution of part of the apparent conflict of particle/wave duality.
Natural Radioactivity of Intrusive-Metamorphic and Sedimentary Rocks of the Balkan Mountain Range (Serbia, Stara Planina)
Stara Planina (also known as the Balkan mountain range) is known for numerous occurrences and deposits of uranium and associated radionuclides. It is also famous for its geodiversity. The geologic framework is highly complex. The mountain is situated between the latitudes of 43° and 44° N and the longitudes from 22°16′ to 23°00′ E. Uranium exploration and radioactivity testing on Stara Planina began back in 1948. Uranium has also been mined in the zone of Kalna, within the Janja granite intrusive. The naturally radioactive geologic units of Stara Planina are presented in detail in this paper. The main sources of radioactivity on Stara Planina can be classified as: 1. Granitic endogenous—syngenetic–epigenetic deposits and occurrences; 2. Metamorphogenic—syngenetic; and 3. Sedimentary, including occurrences of uranium deposition and fluctuation caused by water in different types of sedimentary rocks formed in a continental setting, which could be classified under epigenetic types. The area of Stara Planina with increased radioactivity (higher than 200 cps), measured by airborne gamma spectrometry, is about 380 square kilometers. The highest values of measured radioactivity and uranium grade were obtained from a sample taken from the Mezdreja uranium mine tailing dump, where 226Ra measures 2600 ± 100 Bq/kg and the uranium grade is from 76.54 to 77.65 ppm U. The highest uranium (and lead) concentration, among all samples, is measured in graphitic schist with high concentrations of organic (graphitic) material from the Inovska Series—99.47 ppm U and 107.69 ppm Pb. Thorium related radioactivity is the highest in granite samples from the Janja granite in the vicinity of the Mezdreja granite mine and the Gabrovnica granite mine tailing dump, and it is the same—250 ± 10 Bq/kg for 232Th, while the thorium grade varies from 30.82 to 60.27 ppm Th. In gray siltstones with a small amount of organic material, the highest radioactivity is related to potassium—2080 ± 90 Bq/kg for 40K.