Fractal Electrodynamics: From Super Antennas to Superlattices

  • D. L. Jaggard


We provide a selected overview of fractal electrodynamics by examining the properties of fractal antennas, arrays and apertures; and investigating the nature of electromagnetic wave scattering from fractal surfaces and superlattices.

Fractals offer an ideal blend of order and disorder that can be used in the design of robust antenna arrays with low or moderate side lobes. More recently, fractal geometry has also been used to design individual antenna elements suitable for multiband operation. We also investigate and review the diffraction of electromagnetic waves by fractal apertures.

Finally, we examine and review the scattering of electromagnetic waves from fractal surfaces and fractal superlattices. In each case fractals imprint their distinctive symmetry on interrogating waves. In this way fractal descriptors of an object are embedded in the scattered wave in such a way that they can often be extracted.


Fractal Dimension Antenna Array Fractal Structure Fractal Aperture Diffract Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Jaggard, D.L. (1990): On Fractal Electrodynamics. In Recent Advances in Electromagnetic Theory, H.N. Kritikos and D.L. Jaggard, editors, Springer-Verlag, New York, 183–224.Google Scholar
  2. 2.
    Jaggard, D.L. (1991): Fractal Electrodynamics and Modeling. In Directions in Electromagnetic Wave Modeling, H.L. Bertoni and L.B. Felsen, editors, Plenum Publishing Co., New York, 435–446.Google Scholar
  3. 3.
    Jaggard, D.L. (1995): Fractal Electrodynamics: Wave Interactions with Discretely Self-Similar Structures. In Electromagnetic Symmetry, C. Baum and H. Kritikos, eds., Taylor and Francis Publishers, Washington, D.C., 231–281.Google Scholar
  4. 4.
    Berry, M.V. (1979): Diffractals. J. Phys. A: Math. Gen. 12, 781–797.CrossRefGoogle Scholar
  5. 5.
    Berry, M.V., Blackwell, T.M. (1981): Diffractal Echoes. J. Phys. A: Math. Gen. 14, 3101–3110.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Jakeman, E. (1982): Scattering by a Corrugated Random Surface with Fractal Slope. J. Phys. A 15, L55–L59.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jakeman, E. (1982): Fresnel Scattering by a Corrugated Random Surface with Fractal Slope. J. Opt. Soc. Am. 72, 1034–1041.CrossRefGoogle Scholar
  8. 8.
    Jakeman, E. (1983): Fraunhofer Scattering by a Sub-Fractal Diffuser. Opt. Acta 30, 1207–1212.Google Scholar
  9. 9.
    Jakeman, E. (1986): Scattering by Fractals. In Fractals in Physics, L.Pietronero and E. Tosatti, editors, Elsevier Science Publishers B.V., New York, 55–60.Google Scholar
  10. 10.
    Allain, C, Cloitre, M. (1986): Optical Fourier Transforms of Fractals. In Fractals in Physics, L. Pietronero and E. Tosatti, editors, Elsevier Science Publishers B.V., New York, 61–64.Google Scholar
  11. 11.
    Bale, H.D., Schmidt, P.W. (1984): Small-Angle X-Ray-Scattering Investigation of Submicroscopic Porosity with Fractal Properties. Phys. Rev. Lett. 53, 596–599.CrossRefGoogle Scholar
  12. 12.
    Schaefer, D.W., Martin, J.E., Wiltzius, P., Cannell, D.S. (1984): Fractal Geometry of Colloidal Aggregates. Phys. Rev. Lett. 52, 2371–2374.CrossRefGoogle Scholar
  13. 13.
    Kaye, B.H., Le Blanc, J.E., Abbott, P. (1985): Fractal Description of the Structure Fresh and Eroded Aluminum Shot Fineparticles. Part. Charact. 2, 56–61.CrossRefGoogle Scholar
  14. 14.
    Schmidt, P.W., Dacai, X. (1986): Calculation of the Small-Angle X-Ray and Neutron Scattering From Nonrandom (Regular) Fractals. Phys. Rev. A 33, 560–566.CrossRefGoogle Scholar
  15. 15.
    Rojanski, D., Huppert, D., Bale, H.D., Dacai, X., Schmidt, P.W., Farin, D., Seri-Levy, A., Avnir, D. (1986): Integrated Fractal Analysis of Silica: Adsorption, Electronic Energy Transfer, and Small-Angle X-Ray Scattering. Phys. Rev. Lett. 56, 2505–2508.CrossRefGoogle Scholar
  16. 16.
    Teixeira, T. (1986): Experimental Methods for Studying Fractal Aggregates. In On Growth and Form: Fractal and Non-Fractal Patterns in Physics, H.E. Stanley and N. Ostrowsky, editors, M. Nijhoff, Boston, MA, 145–162.Google Scholar
  17. 17.
    Rarity, J.G., Pusey, P.N. (1986): Light Scattering from Aggregating Systems: Static, Dynamic (QELS) and Number Fluctuations. In H. E. Stanley and N. Ostrowsky, editors, On Growth and Form: Fractal and Non-Fractal Patterns in Physics, Martinus Nijhoff Publishers, Boston, MA, 218–221.Google Scholar
  18. 18.
    Kaye, B.H. (1986): Fractal Dimension and Signature Waveform Characterization of Fine Particle Shape. American Laboratory, 55–63.Google Scholar
  19. 19.
    Chen, Z., Sheng, P., Weitz, D.A., Lindsay, H.M., Lin, M.Y., Meakin, P. (1988): Optical Properties of Aggregate Clusters. Phys. Rev. B37, 5232–5235.Google Scholar
  20. 20.
    Rarity, J.G., Seabrook, R.N., Carr, R.J.G. (1989): Light-Scattering Studies of Aggregation. In Fractals in the Natural Sciences, M. Fleischmann, D.J. Tildesley, and R.C Ball, editors, Princeton University Press, Princeton, NJ, 89–101.Google Scholar
  21. 21.
    Lin, M.Y., Lindsay, H.M., Weitz, D.A., Ball, R.C, Klein, R., Meakin, P. (1989): Universality of Fractal Aggregates as Probed by Light Scattering. In Fractals in the Natural Sciences, M. Fleischmann, D. J. Tildesley, and R. C. Ball, editors, Princeton University Press, Princeton, NJ, 71–87.Google Scholar
  22. 22.
    Sinha, S.K. (1988): Scattering from Fractal Structures. Physica D 38, 310–314.CrossRefGoogle Scholar
  23. 23.
    Kim, Y., Jaggard, D.L. (1986): The Fractal Random Array. Proc. IEEE. 74, 1278–280.CrossRefGoogle Scholar
  24. 24.
    Lakhtakia, A., Varadan, V.K., Varadan, V.V. (1987): Time Harmonic and Time-Dependent Radiation by Bifractal Dipole Arrays. Int. J. Electronics 63, 819–824.CrossRefGoogle Scholar
  25. 25.
    Werner, D.H., Werner, P.L. (1995): On the Synthesis of Fractal Radiation Patterns. Radio Science 30, 29–45.CrossRefGoogle Scholar
  26. 26.
    Puente, C, Pous, R. (1996): Fractal Design of Multiband and Low Side-Lobe Arrays. IEEE Trans. Antennas Propagat. 44, 730–739.CrossRefGoogle Scholar
  27. 27.
    Werner, D.H., Werner, P.L. (1996): Frequency-Independent Featues of Self-Similar Fractal Antennas. Radio Science 31, 1331–1343.CrossRefGoogle Scholar
  28. 28.
    Puente, C. (Dec. 1993): Fractal Design of Multiband Antenna Arrays. Departmenet of Electrical and Computer Engineering, University of Illinois, Urbana-Champaign, ECE 477 Term Project.Google Scholar
  29. 29.
    Cohen, N. (Summer 1995): Fractal Antennas Part 1. Comm. Quart., 7–22.Google Scholar
  30. 30.
    Cohen, N. (Summer 1996): Fractal Antennas Part 2. Comm. Quart., 53–66.Google Scholar
  31. 31.
    Cohen, N., Hohfeld, R.G. (Winter 1996): Fractal Loops and the Small Loop Approximation. Comm. Quart. 77–81.Google Scholar
  32. 32.
    Puente, C, Romeau, J., Pous, R., Garcia, X., Benitez, F. (1996): Fractal Multi-band Antenna Based on the Sierpinski Gasket. IEE Elte. Lett. 32, 1–2.CrossRefGoogle Scholar
  33. 33.
    Puente, C, Romeau, J., Pous, R., Garcia, X., Benitez, F. (1996): Perturbation of the Sierpinski Antenna to Allocate Operating Bands. IEE Elec. Lett. 32, 2186–2187.CrossRefGoogle Scholar
  34. 34.
    Allain, C, Cloitre, M. (1987): Spatial Spectrum of a General Family of Self-Similar Arrays. Phys. Rev. A 36, 5751–5757.MathSciNetCrossRefGoogle Scholar
  35. 35.
    Beal, M.M., George, N. (1989): Features in the Optical Transforms of Serrated Apertures and Disks. J. Opt. Soc. Am A6, 1815–1826.CrossRefGoogle Scholar
  36. 36.
    Jaggard, D.L., Spielman, T., Sun, X¿ (June 24–27, 1991): Diffraction by Cantor Targets”, presented at the 1991 AP-S/URSI Meeting, London, Ontario.Google Scholar
  37. 37.
    Kim, Y., Grebel, H., Jaggard, D.L. (1991): Diffraction by Fractally Serrated Apertures. J. Opt. Soc. A8, 20–26.Google Scholar
  38. 38.
    Spielman, T., Jaggard, D.L. (July 20–25, 1992): Diffraction by Cantor Targets: Theory and Experiment, presented at the 1992 AP-S/URSI Meeting, Chicago, IL.Google Scholar
  39. 39.
    Jaggard, D.L., Spielman, T. (1992): Diffraction from Triadic Cantor Target Diffraction. Microwave and Optical Technology Letters 5, 460–466.CrossRefGoogle Scholar
  40. 40.
    Jaggard, D.L., Spielman, T., Dempsey, M. (June 28–July 2, 1993): Diffraction by Two-Dimensional Cantor Apertures. Presented at the 1993 AP-S/URSI Meeting, Ann Arbor, MI.Google Scholar
  41. 41.
    Jaggard, D.L., Jaggard, A.D. (July 13–18, 1997): Fractal Apertures: The Effect of Lacunarity. to be presented at the 1997 AP-S/URSI Meeting, Montreal.Google Scholar
  42. 42.
    Jaggard, D.L., Sun, X. (1990): Scattering by Fractally Corrugated Surfaces. J. Opt. Soc. A7, 1131–1139.Google Scholar
  43. 43.
    Sun, X., Jaggard, D.L. (1990): Wave Scattering from Non-Random Fractal Surfaces. Opt. Comm. 78, 20–24.CrossRefGoogle Scholar
  44. 44.
    Jaggard, D.L., Sun, X. (1990): Rough Surface Scattering: A Generalized Rayleigh Solution. J. Appl. Phy. 68, 5456–5462.CrossRefGoogle Scholar
  45. 45.
    McScharry, P., Cullen, P.J. (1995): Wave Scattering by a One-Dimensional Band-Limited Fractal Surface Based on a Perturbation of the Green’s Func¬tion. J. Appl. Phy. 78, 6940–6948.CrossRefGoogle Scholar
  46. 46.
    Savaidis, S., Frangos, P., Jaggard, D.L., Hizanidis, K. (Dec. 1995): Scattering from Fractally Corrugated Surfaces: An Exact Approach. Optics Lett. 20, 2357–2359.CrossRefGoogle Scholar
  47. 47.
    Savaidis, S., Frangos, P., Jaggard, D.L., Hizanidis, K. (1997): Scattering from Fractally Corrugated Surfaces Using the Extended Boundary Condition Method. J. Opt. Soc. Am. A14, 475–485.CrossRefGoogle Scholar
  48. 48.
    Jaggard, D.L., Sun, X. (1990): Reflection from Fractal-Layers. Opt Lett. 15, 1428–1430.CrossRefGoogle Scholar
  49. 49.
    Konotop, V.V., Yordanov, O.I., Yurkevich, I.V. (1990): Wave Transmission Through a One-Dimensional Cantor-Like Fractal Medium. Europhys. Lett. 12, 481–485.CrossRefGoogle Scholar
  50. 50.
    Sun, X., Jaggard, D.L. (1991): Wave Interactions with Generalized Cantor Bar Fractal Multi-Layers. J. Appl. Phy. 70, 2500–2507.CrossRefGoogle Scholar
  51. 51.
    Megademini, T., Pardo, B., Jullien, R. (1991): Fourier Transform and Theory of Fractal Multilayer Mirrors. Opt. Comm. 80, 312–316.CrossRefGoogle Scholar
  52. 52.
    Jaggard, D.L., Jaggard, A.D. (1997): Polyadic Cantor Superlattices with Variable Lacunarity. Opt. Lett. 22, 145–147.CrossRefGoogle Scholar
  53. 53.
    Jaggard, A.D., Jaggard, D.L. (July 13–18, 1997): Fractal Superlattices and Scattering: Lacunarity, Fractal Dimension, and Stage of Growth. To be pre¬sented at the 1997 AP-S/URSI Meeting, Montreal; also Jaggard, A.D., Jaggard, D.L. (1997): Scattering from Fractal Superlattices with Variable Lacunarity. In preparation.Google Scholar
  54. 54.
    Mandelbrot, B.B. (1983): The Fractal Geometry of Nature. W. H. Freeman and Comapny, New York.Google Scholar
  55. 55.
    Barnsley, M. (1988): Fractals Everywhere. Academic Press, New York.MATHGoogle Scholar
  56. 56.
    Voss, R. (1985): Random Fractals: Characterization and Measurement. In Scaling Phenomena in Disordered Systems, NATO ASI Series, R. Pynn and A. Skjeltorp, editors, Plenum Press, New York, 1–11.Google Scholar
  57. 57.
    Mandelbrot, B.B. (1994): A Fractal’s Lacunarity, and How it can be Tuned and Measured. In Fractals in Biology and Medicine, T. F. Nonnenmacher, G. A. Losa, and E. R. Weibel, editors, Birkhäuser Verlag, Basel, 8–21.CrossRefGoogle Scholar
  58. 58.
    Mandelbrot, B.B., Stauffer, D. (1994): Antipodal Correlations and the Texture (Fractal Lacunarity). In Critical Percolation Clusters. J. Phys. A: Math. Gen. 27, L237–L242.MathSciNetMATHGoogle Scholar
  59. 59.
    Fabio, D.A., Reis, A., Riera, R. (1994): Lacunarity Calculation in the True Fractal Limit. J. Phys. A: Math. Gen 27, 1827–1835.MATHCrossRefGoogle Scholar
  60. 60.
    Mandelbrot, B.B., Vespignani, A., Kaufman, H. (1995): Crosscut Analysis of Large Radial DLA: Departures from Self-Similarity and Lacunarity Effects. Europhys. Lett 33, 199–204.CrossRefGoogle Scholar
  61. 61.
    Hovi, J.-P., Aharony, A., Stauffer, D., Mandelbrot, B.B. (1996): Gap Independence and Lacuanrity in Percolation Clusters. Phys. Rev. Lett. 77, 877–880.CrossRefGoogle Scholar
  62. 62.
    Schelkunoff, S.A. (1943): A Mathematical Theory of Linear Arrays. Bell Syst. Tech. J. 22, 80–107.MathSciNetMATHGoogle Scholar
  63. 63.
    Lakhtakia, A., Holter, N.S., Messier, R., Varadan, V.K., Varadan, V.V. (1986): On the Spatial Fourier Transforms of the Pascal-Sierpinski Gaskets. J. Phys. A: Math. Gen. 19, 3147–3152.MathSciNetCrossRefGoogle Scholar
  64. 64.
    Uozumi, J., Kimura, H., Asakura, T. (1990): Fraunhofer Diffraction by Koch Fractals. J. Mod. Optics 37, 1011–1031.MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Uozumi, J., Kimura, H., Asakura, T. (1991): Fraunhofer Diffraction by Koch Fractals: The Dimensionality. J. Mod. Optics 38, 1335–1347.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 1997

Authors and Affiliations

  • D. L. Jaggard
    • 1
  1. 1.Complex Media Laboratory, Moore School of Electrical EngineeringUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations