Self-similarity in diffraction by a self-similar fractal screen

Abstract

In the case of Fig. I@), Le., when the edge is illuminated by a creeping mode excited by an earlier surface diffraction, we have n u d ( Q l ) = ui(L) TSc(q1, vl)eiUILMl/a T ( e ikR'1 re 9 1 9 92; V I , *Iud( Q 2 ) = ui(L) Tsc(ql, vl)eivlLM1/u T L (ql, 92; v l , vl)eblMIM2/" n eikI?2 . Ta(919 V I ) T a n ud(Q3)=ui(L)Tsc(ql, vl)eivlLw1/aTA2)(qI, q2; v l , pl)ei~~~w11w3'u n eikh?3 * Tm(12, P I ) -In the above expressions ui stands for the incident field, while the meanings of various geometrical parameters are shown in Fig. 1. p1 is the first zero of q2Ht1)(x) -iH;(l)(x) and the explicit expressions for the diffraction coefficients T, are found as listed below When a -t 03, the edge diffraction coefficient, is reduced to which is merely the coefficient related to the two-part impedance half-plane. REFERENCESClemmov, The Plane Wave Spectrum Representation of Electromagnetic Fields. Elmsford, N Y : Pergamon, 1966. R. Tiberio and G . Pelosi, "High-frequency scattering from the edges of impedance discontinuities on a flat plane," IEEE Tram. Antennas Propagat., vol. AP-31, no. 4, pp. 590-596, 1983. M. Idemen and L. B. Felsen, "Diffraction of a whispering gallery mode by the edge of a thin concave cylindrically curved surface," IEEE Trans. Antennas Propagat., vol. A€-29, pp. 571-579, 1981. T. B. A. Senior, "Half plane edge diffraction," Radio Sei., vol. 10, no. 6, pp. 645-650, 1975. A. Biiyiikaksoy and G . Uzgoren, "High frequency scattering by a cylindrically curved impedance surface," AEU, bd. 39, no. 6, pp. 351-358, 1983. M. Idemen, "Necessary and,sufficient conditions for a surface to be. an impedance boundary," AEU, bd. 35, no. 2, pp. 84-86, 1981. M. Idemen and E. Erdogan, "Diffraction of a creeping wave generated on a perfectly conducting spherical scatterer by a ring source," ZEEE Trans.Abstract-Using the Sierpinski gasket as an example, it has been shown here that the paraxial Fraunhoffer-zone diffracted field of a self-similar fractal screen also exhibits self-similarity. This also establishes that fractal structures can be used with great profit in problems involving array syntheses. Recently, it bas been shown that the Sierpinski gasket is itself a member of a much wider class of gaskets, and the potential for the use of fractal structures in electromagnetic (EM) problems may he vast indeed.

Time-harmonic and time-dependent radiation by bifractal dipole arrays

“…The parameter N > 1 for the gaskets chosen here is a prime integer, while the initiator f(x; a) as well as the generator g(x; a) (Mandelbrot 1983) are given by the function ) in which the spatial frequencies used are u = wxo/ro c, v = wyo/r o c. Rewriting eqn (10) in the form (12) we immediately observe that qL(X O, t) at any given X o is a unifractal in time because the relationships (13 a,b) bear a strong analogy to eqns (9 a.b), Thus, it is seen that a dipole array which is bifractal in its spatial arrangement radiates a far-field which is also a bifractal in its spatial variations (Lakhtakia et al 1987). In addition, the time-dependent far-field response of this array turns out to a unifractal in time.…”

“…Since we have modelled these gaskets as collections of nodes (Lakhtakia et at. 1986(Lakhtakia et at. , 1987 and not as collections of triangles in the usual manner (Mandelbrot 1983) of modelling the Sierpinski gasket, the concept behind the Pascal-Sierpinski gaskets can be extended to yield a simple procedure for constructing fractal arrays of antennas.…”

Analysis of Nonlinear Fractal Optical Antenna Arrays - A Conceptual Approach

“…Fractal theory is an emerging theory which revolutionized the way the scientists think about the nature of the world [2][3][4][5][6][7][8][9][10]. Derived from the Latin word meaning break apart the term fractal was originally coined by Mandelbrot to describe a family of complex shapes that possess an inherent self-similarity or self-affinity in their geometrical structure.…”

Analysis of Novel Fractal Optical Antenna Arrays - A Conceptual Approach

“…FRACTAL THEORY is a relatively emerging field of mathematics that has changed the way scientists view of looking at a natural occurring phenomena in this world [1][2][3][4][5][6][7][8][9]. This paper makes an attempt of extending the concept of fractal antenna array to the optical regime for the design of fractal optical antenna arrays.…”