A Brief Historical Review

The fractal design of antennas and arrays results from the blend of two aparently disjoint disciplines: electromagnetic theory and geometry. From the early spiral and log-periodic antennas developed in the early sixties by Carrel, Mayes et al, and from the works of Benoit Mandelbrot on fractal geometry, the fractal design of antennas appears as a natural way to explore for a multifrequency operation and for an antenna size reduction. This document reviews the development of such a novel concept and gives the reader some references for further reading.

Background

• 40's. L.J.Chu and H.Wheeler stablish a fundamental limit on the performance of small antennas.

• 60's. Several research teams at the University of Illinois at Urbana-Champaign, University of California and Ohio State University introduce the concept of frequency-independent antennas. The self-scaling concept gives place to the log-periodic dipole array and to the spiral antennas.

• 70's. B. Mandelbrot coins the term 'Fractal'. Such a name is used to describe a family of geometrical objects that defy the traditional rules of the Euclidean geometry. Later on, Mandelbrot states that such a weird fractal shapes are, in fact, among the most common forms in nature.

• 80's. D.Jaggard et al. coin the term 'Fractal Electrodynamics'. Electromagnetics and Fractal Geometry blend into a novel discipline. The interaction of fractal bodies and electromagnetic waves are investigated and some relationships between common fractal properties and the scattered electromagnetic waves are stablished.

• 90's. The first reported fractal multiband antennas are introduced by C.Puente et al. from the Universitat Politècnica de Catalunya. Also, the potentiallity of fractals to become small antennas is introduced by the same research team and by N.Cohen from the University of Boston.

Some Significant Milestones

• 1986-1993. Several authors theoretically investigate  the relationships between fractal arrays and their array factors. Some fractal and array properties such as  Fractal Dimension and Secondary Lobe Ratio are linked for the first time.

• 1993, November. The potentiallity of fractal arrays to become multifrequency arrays is introduced by C.Puente at the University of Illinois at Urbana-Champaign during a graduate course discussion.

• 1994. Some prelimar ideas on multiband fractal arrays are presented by C.Puente at an URSI meeting in Las Palmas, Gran Canaria, Spain. Further work on such ideas gives place to a paper that is submitted to the IEEE Transactions on Antennas and Propagation in May 1994, and published in May 1996.

• 1995, May. After several months working on the invention of the Sierpinski antenna, the Universitat Politècnica de Catalunya finally applies for the patent on Fractal and Multifractal Antennas . Such results proved the feasibility of fractals to become multiband antennas (i.e., antennas that keep the same parameters at several frequency bands).  A paper reporting the first results is submitted to IEE Electronics Letters.

• 1995, Summer. Some preliminar ideas on fractal small antennas are suggested by N.Cohen from the Boston University.

• 1996-1998. The first reported experimental results on fractal multiband antennas are published by the Electromagnetics & Photonics Engineering research team from the Universitat Politècnica de Catalunya.

• 1998. The Koch monopole becomes the first reported fractal small antenna that improves the features of some classical antennas in terms of bandwidth, resonance frequency and radiation resistance. Whether such antennas are bounded to the fundamental limits on small antennas is still a topic under investigation.

• 1998. The Fractal Team from the Electromagnetics & Photonics Engineering group (UPC), in cooperation with Fractus^TM the Fractal Antennas^TM company from the Sistemas Radiantes F.Moyano S.A. group, develope the first commercial multiband fractal shape antennas for GSM + DCS cellular phone systems.

References

Fractal Shape Antennas

1. C.Puente, R.Pous, J.Romeu, X.García, "Antenas Fractales o Multifractales", Invention Patent, nº: P-9501019. Presented at the Oficina Española de Patentes y Marcas. Owner: Universitat Politècnica de Catalunya, May 1995.
2. C.Puente,J.Romeu,R.Pous,X.Garcia, F.Benítez, “Fractal Multiband Antenna Based on the Sierpinski Gasket”, IEE Electronics Letters, vol.32, no.1, pp.1-2, January 1996.
3. C.Puente, R.Pous, "Fractal Design of Multiband and Low Side-Lobe Arrays", IEEE Transactions on Antennas and Propagation, vol.44, no.5, pp.730-739, May 1996.
4. C.Puente,J.Romeu, R.Bartolomé, R.Pous, “Perturbation of the Sierpinski antenna to allocate the operating bands”, IEE Electronics Letters, vol. 32, no.24, pp. 2186-2188, November 1996.
5. C.Puente, J.Claret, F.Sagués,J.Romeu, M.Q.Lopez-Salvans, R.Pous, “Multiband properties of a fractal tree antenna generated by electrochemical deposition,” IEE Electronics Letters, vol.32, no.5, pp. 2298-2299, December 1996.
6. C.Puente, J.Romeu, R.Pous, A.Cardama, “Multiband Fractal Antennas and Arrays”, in Fractals in Engineering, J.L.Véhel, E.Lutton, C.Tricot editors, Springer, New York 1997.
7. C.Borja, C.Puente, A.Medina, J.Romeu, R.Pous, “Modelo Sencillo para el Estudio de los Parámetros de Entrada de una Antena Fractal de Sierpinski”, XII Simposium Nacional URSI, vol.I, pp.363-371, Bilbao, Septiembre 1997.
8. C.Puente, M.Navarro, J.Romeu, R.Pous, “Efecto de la Variación Angular del Vértice de Alimentación en la Antena Fractal de Sierpinski”, XII Simposium Nacional URSI, vol.I, pp.363-371, Bilbao, Septiembre 1997.
9. M.Navarro, C.Puente, R.Bartolomé, A.Medina, J.Romeu, R.Pous, “Modificación de la Antena de Sierpinski para el Ajuste de las Bandas Operativas”, XII Simposium Nacional URSI, vol.I, pp.363-371, Bilbao, Septiembre 1997.
10. C.Puente, J.Romeu, R.Pous, A.Cardama, “Multiband Fractal Antennas and Arrays”, Fractals in Engineering Conference, INRIA Rocquencourt, Arcachon, France, June 1997.
11. C.Puente, "Fractal Antennas", Ph.D. Dissertation at the Dept. of Signal Theory and Communications, Universitat Politècnica de Catalunya, June 1997.
12. C.Puente, J.Romeu, R.Pous, J.Ramis, A.Hijazo, "Small but long Koch fractal Monopole", IEE Electronics Letters, vol.34, no.1, pp.9-10, January, 1998.
13. C.Puente, J.Romeu,R.Pous,A.Cardama, “On the Behavior of the Sierpinski Multiband Fractal Antenna”, IEEE Trans. on Antennas & Propagation, April 1998.
14. C.Puente, M.Navarro, J.Romeu, R.Pous, “Variations on the Fractal Sierpinski Antenna Flare Angle", IEEE Antennas & Propagation - URSI Symposium Meeting, Atlanta, June 1998.
15. N.Cohen, "Fractal Antennas: Part 1", Communications Quarterly, pp.7-22, Summer 1995.
16. N.Cohen, R.G.Hohlfeld, "Fractal Loops And The Small Loop Approximation", Communications Quarterly, pp.77-81, Winter 1996.
17. N.Cohen, "Fractal and Shaped Dipoles", Communications Quarterly, pp. 25-36, Spring 1996.
18. N.Cohen, "Fractal Antennas: Part 2", Communications Quarterly, pp. 53-66, Summer 1996.
19. 
     

Fractal Arrays

1. Y.Kim and D.L.Jaggard, "The Fractal Random Array", Proc. of the IEEE, vol.74, no. 9, pp.1278-1280, September 1986.
2. C.Puente, "Fractal Design of Multiband Antenna Arrays", Elec. Eng. Dept. Univ. Illinois, Urbana-Champaign, ECE 477 term project, Dec.1993.
3. C.Puente, R.Pous, "Diseño Fractal de Agrupaciones de Antenas", IX Simposium Nacional URSI, vol.I, pp. 227-231, Las Palmas, Septiembre 1994.
4. C.Puente, R.Pous, "Fractal Design of Multiband and Low Side-Lobe Arrays", IEEE Transactions on Antennas and Propagation, vol.44, no.5, pp.730-739, May 1996.
5. C.Goutelard, "Fractal theory of large arrays of lacunar antennas," Electromagnetic Wave Propagation Panel Symposium (AGARD-CP-528), pp. 35/1-15, June 1992 (in french).
6. A.Lakhtakia, V.K. Varadan, V.V. Varadan, "Time-harmonic and time-dependent radiation by bifractal dipole arrays", Int. J. Electronics, 1987, vol. 63, no. 6, pp.819-824, 1987.
7. C.Allain, M.Cloitre, "Spatial spectrum of a general family of self-similar arrays", Physical Review A, vol.36, no.12, pp.5751-5757, December 1987.
8. X.Liang, W. Zhensen, W.Wenbing, "On the synthesis of fractal patterns from the concentric-ring array," IEE Electronics Letters, vol.32, pp.1940-1941, October 1996.
9. D.H. Werner and P.L. Werner, “On the synthesis of fractal radiation patterns,” Radio Science, Vol. 30, No. 1, pp.29-45, Jan.-Feb. 1995.
10. D.H. Werner and R.L. Haupt, “Fractal constructions of linear and planar arrays,” 1997 IEEE Antennas and Propagation Society Symposium Digest, Montreal, Canada, July 1997, pp.1968-1971.
11. D.H.Werner, P.L.Werner, "Frequency-independent features of self-similar fractal antennas", Radio Science, Volume 31, No.6,pp.1331-1343, November-December 1996.

Fractal Electrodynamics

1. A.Lakhtakia, N.S. Holter, R. Messier, V.K.Varadan, V.V. Varadan, "On the Spatial Fourier Transforms of the Pascal-Sierpinski Gaskets", J.Phys.A: Math.Gen. 19, pp.3147-3152, 1986.
2. D.L.Jaggard, "On Fractal Electrodynamics," in Recent Advances in Electromagnetic Theory, H.N.Kritikos, pp. 183-224, D.L.Jaggard Editors; Springer Verlag, 1990.
3. D.L.Jaggard, "Prolog to special section on Fractals in Electrical Engineering," Proceedings of the IEEE, vol. 81, no. 10, pp. 1423-1427, October 1993.
4. Y.Kim and D.L.Jaggard, "The Fractal Random Array", Proc. of the IEEE, vol.74, no. 9, pp.1278-1280, September 1986.
5. D.L.Jaggard and Y.Kim, "Diffraction by Bandlimited Fractal Screens," J.Opt.Soc.Am.A4, 1055-1062 (1987).
6. Y.Kim and D.L.Jaggard, "A Bandlimited Fractal Model of Atmospheric Refractivity Fluctuation," J.Opt.Soc.Am.A5, pp. 475-480, (1988).
7. Y.Kim and D.L.Jaggard, "Optical Beam Propagation in a Bandlimited Fractal Medium," J.Opt.Soc.Am.A5, pp. 1419-1426, (1988).
8. D.L.Jaggard, X.Sun, "Scattering from Bandlimited Fractal Fibers," IEEE Trans. Ant. and Propagat., vol.37, pp.1591-1597 (1989).
9. D.L.Jaggard, X.Sun, "Scattering by Fractally Corrugated Surfaces,"J.Opt.Soc.A7, pp.1131-1139, (1990).
10. X.Sun and D.L.Jaggard, "Wave Scattering from Non-Random Fractal Surfaces" Opt.Comm. 78, pp.20-24, (1990).
11. D.L.Jaggard, X.Sun, "Rough Surface Scattering: A Generalized Rayleigh Solution", J.Appl.Phys. 68, pp.5456-5462, December 1990.
12. D.L.Jaggard, X.Sun, "Reflection from Fractal-Layers," Optics Let. 15, pp. 1428-1430 (1990).
13. D.L. Jaggard and T.Spielman, "Triadic Cantor Target Diffraction," Microwave and Optical Technology Letters, vol. 5, no. 9, pp. 460-466, August 1992.
14. D.L.Jaggard, T.Spielman, X.Sun, "Fractal Electrodynamics and Diffraction by Cantor Targets", AP-S/URSI Comission B Meeting, London, Ontario, June 1991.
15. X.Sun, D.L.Jaggard, "Wave interactions with generalized Cantor bar fractal multilayers," J. Appl. Physics, 70 (5), pp. 2500-25007, September 1991.
16. D.L.Jaggard, X.Sun, "Reflection from Fractal Multilayers", Optics Letters, vol.15, no.24, December 1990.
17. M.M.Beal, N.George, "Features in the optical transforms of serrated apertures and disks", J. Opt.Soc.Am., vol. 6, no. 12, pp. 1815-1826, December 1989.
18. Y. Kim, H. Grebel, D.L. Jaggard,"Diffraction by fractally serrated apertures", J.Opt.Soc.Am., vol. 8, no. 1, pp. 20-26, January 1991.
19. D.L.Jaggard, "Fractal Electrodynamics: Wave Interactions with Discretely Self-Similar Structures," chapter 5 in Electromagnetic Symmetry, C.Baum and H.Kritikos, eds., Taylor and Francis Publishers, 1995.
20. M.V.Berry, "Diffractals", J.Phys.A: Math.Gen.12, pp.781-797, (1979).
21. M.V.Berry, T.M.Blackwell, "Diffractal Echoes", J.Phys.A:Math.Gen.14, pp.3101-3110, 1981.
22. C.Allain, M.Cloitre, "Optical Fourier Transforms of Fractals," in Fractal Physics, L.Pietronero and E.Tossatti, editors, North-Holland, New York, 1986.
23. S.D.Bedrosian, X.Sun, "Theory and Application of Pascal-Sierpinski Gasket Fractals", Circuits Syst., Sig.Processing 9, pp.147-159, (1990).
24. S.D.Bedrosian, X.Sun, "Pascal-Sierpinski Gasket Fractal Networks: Some Resistance Properties", J.Franklin Inst.326, pp.503-509, (1989).
25. T.Falco,F.Francis,S.Lovejoy,D.Schertzer, B.Kerman, M.Drinkwater, "Universal Multifractal Scaling of Synthetic Aperture Radar Images of Sea-Ice", IEEE Transactions on Geoscience and Remote Sensing, vol.34, no.4, July 1996.
26. C.V.Stewart, B.Moghaddam, K.J.Hintz and L.M.Novak, "Fractional Brownina Motion Models for Synthetic Aperture Radar Imagery Scene Segmentation," Proc. IEEE, vol.81, pp. 1511-1522, 1993.

Fractals

1. B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, 1983.
2. H.O. Peitgen, H.Jürgens, D.Saupe, Chaos and Fractals, New Frontiers of Science, Springer-Verlag, 1990.
3. B.B.Mandelbrot,Los Objetos Fractales, Tusquets Editores, Barcelona, 1975.
4. H.Koch, "Sur une courbe continue sans tangente, obtenue par une construction géometrique élémentaire," Arkiv för Matematik 1, pp.681-704,1904.
5. M.F. Barnsley , R.L. Devaney, B.B. Mandelbrot, H.O. Peitgen, D. Saupe, R.F. Voss, Y. Fisher, M. Mc Guire, The Science of Fractal Images, Springer Verlag, 1988.
6. W.Sierpinski, "Sur une courbe dont tout point est un point de ramification", C.R.Acad. Paris 160, p.302,1915.
7. K.Falconer, Fractal Geometry, Mathematical Foundations and Applications, Wiley, NewYork, 1990.
8. H. Jones, D.E. Reeve, D. Saupe, Fractals and Chaos, A.J. Crilly, R.A. Earnshaw and H. Jones, Editors; Springer-Verlag, 1990.
9. L.López-Tomàs, J.Claret, F.Mas, F. Sagués, "Aggregation under a forced convective flow", Physical Review B, vol. 46, no.18, pp. 11 495-11 500, November 1992.
10. F.Sagués, L.López-Tomàs, J.March, R.Reigada, P.P.Trigueros, E.Vilaseca, J.Claret, F.Mas, "Disordered Grown Systems: Generation and Fractal Analysis. Electrodeposition", Int. Journal of Quantum Chemistry, vol.52, pp.375-394,1994.
11. L.López-Tomàs, J.Claret, F.Sagués, "Quasi Two-Dimensional Electrodeposition under Forced Fluid Flow", Physical Review Letters, vol. 71, no. 26, pp.4373-4376, December 1993.
12. R.F.Voss, "Multiparticle diffusive fractal aggregation", Phyiscal Review B, vol.30, no.1, pp.334-337, July 1984.

Frequency Independent and Small Antennas

1. P.E.Mayes, "Frequency-Independent Antennas and Broad-Band Derivatives Thereof", Proc. of the IEEE, vol. 80, no. 1, January 1992.
2. J.D.Dyson, "The Equiangular Spiral Antenna," IRE Trans. Antennas Propag., vol. AP-7, pp. 181-187, October 1959.
3. J.D.Dyson, "The Unidrectional Equiangular Spiral Antenna," IRE Trans. Antennas Propag., vol. AP-7, pp. 329-334, October 1959.
4. D.E.Isbell, "Log-Periodic Dipole Arrays, " IRE Trans. Antennas Propagation, vol.8, pp. 260-267, May 1960.
5. P.E.Mayes, G.A. Deschamps, and W.T. Patton, "Backward-wave radiation from periodic structures and application to the design of frequency-independent antennas," Proc. IRE, vol. 49, pp. 962-963, May 1961.
6. G.A. Deschamps, J.D. Dyson, "The Logarithmic Spiral in a Single-Aperture Multimode Antenna System," IEEE Transactions on Antennas and Propagation, vol. ap-19, no. 1, January 1971.
7. V.H.Rumsey, Frequency-Independent antennas, Academic Press, 1966.
8. R.L. Carrel, "Analysis and Design of the log-periodic dipole antenna," Doctoral Thesis at the Dep. of Electrical Engineering of the University of Illinois at Urbana-Champaign, 1961.
9. R.S.Elliot, "A View of Frequency-Independent Antennas," The Microwave Journal, pp.61-68, December 1962.
10. P.E.Mayes, Advanced Antenna Theory, ECE 477 course, Dept. of Electrical Engineering, University of Illinois, Urbana-Champaign, 1993.
11. H.G.Booker, "Slot Aerials and Their Relation to Complementary Wire Aerials," J.Inst.Elec.Engrs., pt III A, pp.620-626, 1946.
12. H.A.Wheeler, "Fundamental Limitations of Small Antennas," Proc. IRE, pp. 1479-1488. December, 1947.
13. R.C.Hansen, "Fundamental Limitations in Antennas", Proc. IEEE, vol.69, no.2, February 1981.
14. G.Goubau, "Multi-Element Monopole Antennas", Proc. Workshop on Electrically Small Antennas ECOM, Ft. Mammouth, N.J., pp.63-67, May 1976.
15. H.A.Wheeler, "Small Antennas", IEEE Transactions on Antennas and Propagation, vol. 23, no.4,pp. 462-469, July 1975.
16. J.S. McLean, "A Re-Examination of the Fundamental Limits on the Radiation Q of Electrically Small Antennas", IEEE Trans. on Antennas and Propagation, Vol.44, no.5, May 1996.
17. C.H.Walter, "Electrically Small Antenna Studies at OSU", Proc. Workshop on Ellectrically Small Antennas ECOM, Ft. Mammouth, N.J., pp.25-34, May 1976.
18. K.Fujimoto, A.Henderson, K.Hirasawa, J.R.James, Small Antennas, Research Studies Press Ltd., West Sussex, February 1988.
19. L.J.Chu, "Physical Limitations on omni-directional antennas", J.Appl.Phys., vol. 19, pp.1163-1175, December 1948.

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