Frequency and Spatial Features of Waves Scattering on Fractals

Transcription

1 nd Chaotic Modeling and Simulation International Conference, -5 June 009, Chania Crete Greece Frequency and Spatial Features of Waves Scattering on Fractals A.V. Laktyunkin, A.A. Potapov V.A. Kotelinikov Institute of Radio Engineering and Electronics RAS, Russia, Moscow, str. Mohovaya,, Abstract: Now there are two general approaches of scattering on the statistically rough surface: method of small perturbation and Kirchhoff approach. These methods relate to the two extreme cases of very small flat irregularities or smooth and large irregularities respectively. We use fractal model which has many advantages. Electrical field dependencies on fractal dimension D are presented. Investigation of the coherence function with accounting of surfaces fractality has been started. Keywords: Fractals, Radio waves scattering, Scattering indicatrixes, Coherence function.. Introduction There are a lot of scientific and engineering problems, which can be successfully solved only with deep understanding of wave scattering characteristics for statistically rough surface []. Thus before the present diffraction problems for the statistically rough surfaces took into account irregularities of only single scale. Soon it had been realized that multi-scale surfaces lead to better fitting. Now by taking into account results of world works we can state that physical meaning of diffraction theory, which includes multi-scale surfaces, becomes clearer with fractal approach and choosing of fractal dimension (fractal signature) D as a parameter [], [3]. Moreover as we have found out fractality accounting makes theoretical and experimental scattering patterns for earth cover closer. This fact is always interpreted as result of instrumental error. In many works it has been shown that diffraction by fractal surfaces fundamentally differs from diffraction by conventional random surfaces and some of classical statistical parameters like correlation length and root-meansquare deviation go to infinity. This fact is result of self-similarity of fractal surface. In our work band-limited Weierstrass function was used. For the scattered field analysis we use Kirchhoff approach [4].. Fractal Surface The most convenient function which both describes fractals well and is easy for using in calculations is modified D band-limited Weierstrass function. It has a view:

2 Laktyunkin A.V., Potapov A.A. N M ( D 3) n n πm πm W ( x, y) = cw q sin Kq x cos( ) + y sin( ) + φnm n= 0 m= M M () where c w - the constant, that provides unit normalization; q > - the fundamental spatial frequency; D the fractal dimension ( < D < 3); K is the fundamental wave number; N and M number of tones; φ nm - an arbitrary phase that has a uniform distribution over the interval [ π, π ]. f f f a b c Figure. W(x,y) for (a) - N =, M = 3, D =.0, q =.0; (b) - N = 5, M = 5, D =.5, q = 3; (c) - N = 0, M = 0, D =.99, q = 7 Since the natural surfaces are neither purely random nor periodical and are often anisotropic [], [4] then function that was proposed above is a good candidate for characterizing of natural surfaces. Figure shows us examples of band-limited Weierstrass function for different scales. It is also important that function () describes the mathematical fractals only if M and N go to infinity. It is clear from figure that the function proposed possesses the self-similarity and multi-scale. 3. Correlation Parameters In this section of our work a statistical parameter is introduced for estimation of fractal dimension D influence and other fractal parameters influence on the surface roughness. Such parameter as correlation length Γ is conventionally used for numerical characterization of rough surface: ( D 3) N ~ ( q ) ( D 3) n n ( τ ) ρ( τ ) q J 0 ( Kq τ ) s ( D 3) N Γ = = () ( q ) n= 0 There are Γ ~ dependences on q and D in figures and 3 respectively. It is shown that with increased value of D, Γ ~ decreases more rapidly for the same variation of q. It is shown in figure that value of Γ ~ reduces steadily with the increase of D value. However Γ ~ does not change when q =,0.

3 Frequency and Spatial Features of Waves Scattering on Fractals - Figurre. Function Γ ~ dependence on D for various values of q Figure 3. Function Γ ~ dependence on q for various values of D 4. Scattering Model and Indicatrixes As mentioned above Kirchhoff approach has been already used for analysis of wave scattering by fractal surfaces [3], [4]. Conventional conditions of Kirchhoff approach applicability are the following: irregularities are large-scale; irregularities are smooth and flat. In the following calculations we assume that observation is carried out from Fraunhofer zone, incident wave is plane and monochromatic, there are no points with infinite gradient on the surface, Fresnel coefficient V 0 is constant for this surface, surface large scales are much greater than incident wave length. Shading effects will be taken into account in the following our investigations and studies. Scattering indicatrix for average field intensity and two-dimensional surface [3], [4]: F g sinc ( θ, θ, θ 3 ) ( kcσ ) cos θ ka + Kq n π m cos L M x sinc sinc (kal ) sinc x kb + Kq n ( kbl y ) + C 4 π m sin L M y N M f n= 0 m = q ( D 3) n (3) 5. Scattered Field and Coherence Function We have got a data base of scattering indicatrixes for various fractal scattering surfaces. Also in terms of Weierstrass function () for onedimensional fractal scattering surface we obtained scattering field absolute value dependences on incident angle and surface fractal dimension D (see figures 4-6). Scattered field E s in one-dimensional case has a view: 3

4 E Laktyunkin A.V., Potapov A.A. / [ sin( α ) ξ ( x) cos( α )] exp( i[ k sin( α ) x k cos( α ) ξ ( x) ] dx l S = x ) (4) l In terms of equation (4) we performed calculation of the field E s for fractal surfaces. It has been shown on figures 4-6 that the higher fractal dimension the higher absolute value of scattered field in both cases. This phenomenon can be explained by growing contribution of secondary scattering on small irregularities as compared to less rough surface. When incident angle increases scattered field changes spontaneously that is result of chaotic structure of the scattering surface. 3,5 3,0 α = 5 degrees M =, N = q =.7 λ =. mm 0,4 D =. M =, N = q =.7 λ =. mm,5 0,3,0 E ( D ),5 E ( α ) 0,,0 0, 0,5 0,0 0,0-0,5,0,,4,6,8,0 D α, degrees Figure 4 Dependence of scattered field absolute value on D ( λ =. mm ) 0, 0,0 Figure 5. Dependence of scattered field absolute value on the incident angle θ = 5 grad M= N= q=.7 λ = 3 cm 0,08 E ( D ) 0,06 0,04 0,0 0,00-0,0,0,,4,6,8,0 Figure 6. Dependence of scattered field absolute value on D for D λ = 30 mm Based on the Kirchhoff approximation of the rough surfaces with small slopes, the two-frequency mutual coherence function is found to be [, 5]: 4

5 Frequency and Spatial Features of Waves Scattering on Fractals - Γ( ω, ω ) = < T( ω) T = A dx [ < exp( iv r i + v r ) > (5) < exp( iv r ) >< exp( i )] dx v r Below on figures 7-8 there are examples of our numerical modeling of frequency coherence function for fractal surface remote sensing. Ψ k (z) Ψ k (z),0,0 0,8 0,8 Coherence function 0,6 0,4 Coherence function 0,6 0, 0,4 0, Height of the observation point, z (m) 0, Observation point height (m) Figure 7. Dependence of the coherence function on observation height on low heights. Figure 8. Dependence of the coherence function on observation height on middle heights. 6. Conclusions The data presented find widespread application in theory and practice of radar and also in new informational technologies. Further investigations of electromagnetic waves scattering on fractals will be proceeded in the framework of coherence frequency function f calculating for radar fractal sensing channel. It can be assumed that, with application of the formalism of dissipative systems (fractal properties, scaling, fractional operators, non- Gaussian statistics, the chaos mode, existence of strange attractors and the topology of such attractors, etc.), the classical problem of wave propagation and scattering by random media will remain a field of productive research. Application of the term fractal in radio physics and radio electronics is not only justified but also necessary. This application requires fundamental changes in the commonly accepted general concepts used by scientists and engineers. References [] F.G. Bass, I.M. Fuks, Wave Scattering from Statistically Rough Surfaces, Oxford, Pergamon Press, 978. [] A.A. Potapov, Fractals in Radio Physics and Radar: Topology of Sample, -th issue ed. and correct, Moscow, University Library, p. 848, 005. k 5

6 Laktyunkin A.V., Potapov A.A. [3] N. Lin, H.P. Lee, S.P. Lim, K.S. Lee, Wave Scattering from Fractal Surfaces, Journal of Modern Optics, vol. 4, No, pp. 5 4, 995. [4] A.A. Potapov, A.V. Laktyunkin, Theory of the Wave Scattering by Anisotropic Fractal Surface, Nonlinear world, vol. 6,, 008, pp [5] L. Guo, C. Kim, Study on the Two-Frequency Scattering Cross Section and Pulse Broadening of the One-Dimensional Fractal Sea Surface at Millimeter Wave Frequency, Progress In Electromagnetics Research, PIER 37, 34, 00 6