volume | PIER Journals

Abstract

In this paper, reflection and transmission coefficients at dielectric fractal-fractal interface are discussed. The ratio of permittivity of the two dielectric fractal media is kept constant, while the dimension is varied in order to get the desired results. Conventional results are recovered for the integer dimensions. The proposed expressions are useful to study the behavior of electromagnetic waves for non-integer dimensions, multiple fractal interfaces and waveguides. Moreover, it is also helpful to understand the variation in the magnitudes of reflection and transmission coefficients with the difference in dimensionality at interface of the two fractal media.

1. Mandelbro, B., The Fractal Geometry of Nature, W. H. Freeman, 1983.

2. Vicsek, T., "Fractal models for diffusion controlled aggregation," J. Phys. A: Math. Gen., Vol. 16, No. 17, 1983.
doi:10.1088/0305-4470/16/17/003 Google Scholar

3. Wagner, G. C., J. T. Colvin, J. P. Allen, and H. J. Stapleton, "Fractal models of protein structure, dynamics and magnetic relaxation," J. Am. Chem. Soc., Vol. 107, No. 20, 5589-5594, 1985.
doi:10.1021/ja00306a001 Google Scholar

4. Stillinger, F. H., "Axiomatic basis for spaces with noninteger dimension," J. Math. Phys., Vol. 18, No. 6, 1224-1234, 1977.
doi:10.1063/1.523395 Google Scholar

5. Bollini, C. G. and J. J. Giambiagi, "Dimensional renormalization: The number of dimensions as a regularizing parameter," Nuovo Cimento B, Vol. 12, 20-26, 1972. Google Scholar

6. Baleanu, D. and S. Muslih, "Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives," Phys. Scripta, Vol. 72, No. 23, 119-121, 2005.
doi:10.1238/Physica.Regular.072a00119 Google Scholar

7. Tarasov, V. E., "Electromagnetic fields on fractals," Modern Phys. Lett. A, Vol. 21, No. 20, 1587-1600, 2006.
doi:10.1142/S0217732306020974 Google Scholar

8. Palmer, C. and P. N. Stavrinou, "Equations of motion in a non-integer-dimensional space," J. Phys. A, Vol. 37, 6987-7003, 2004.
doi:10.1088/0305-4470/37/27/009 Google Scholar

9. Tarasov, V. E., "Continuous medium model for fractal media," Physics Letters A, Vol. 336, No. 2-3, 167-174, 2005.
doi:10.1016/j.physleta.2005.01.024 Google Scholar

10. Martin, O.-S., "Electromagnetism on anisotropic fractals,", 2011,Eprint arXiv: 1106.1491. Google Scholar

11. Muslih, S. and D. Baleanu, "Fractional multipoles in fractional space," Nonlinear Analysis: Real World Applications, Vol. 8, 198-203, 2007.
doi:10.1016/j.nonrwa.2005.07.001 Google Scholar

12. Baleanu, D. and A. K. Golmankhaneh, "On electromagnetic field in fractional space," Nonlinear Analysis: Real World Applications, Vol. 11, No. 1, 288-292, 2010.
doi:10.1016/j.nonrwa.2008.10.058 Google Scholar

13. Zubair, M., M. J. Mughal, Q. A. Naqvi, and A. A. Rizvi, "Differential electromagnetic equations in fractional space," Progress In Electromagnetic Research, Vol. 114, 255-269, 2011. Google Scholar

14. Zubair, M., M. J. Mughal, and Q. A. Naqvi, "The wave equation and general plane wave solutions in fractional space," Progress In Electromagnetics Research Letters, Vol. 19, 137-146, 2010. Google Scholar

16. Zubair, M., M. J. Mughal, and Q. A. Naqvi, "An exact solution of the cylindrical wave equation for electromagnetic field in fractional dimensional space," Progress In Electromagnetics Research, Vol. 114, 443-455, 2011. Google Scholar

17. Zubair, M., M. J. Mughal, and Q. A. Naqvi, "An exact solution of spherical wave in D-dimensional fractional space," Journal of Electromagnetic Waves and Applications, Vol. 25, No. 10, 1481-1491, 2011. Google Scholar

18. Zubair, M., M. J. Mughal, and Q. A. Naqvi, "Electromagnetic fields and waves in fractional dimensional space," Springer Briefs in Applied Sciences and Technology, XII, 76, Springer, Germany, Jan. 28, 2012. Google Scholar

19. Mughal, M. J. and M. Zubair, "Fractional space solutions of antenna radiation problems: An application to Hertzian dipole," 2011 IEEE 19th Conference on Signal Processing and Communications Applications (SIU), 62-65, Apr. 20-22, 2011, doi:10.1109/SIU.2011.5929587. Google Scholar

20. Hira, A., M. Zubair, and M. J. Mughal, "Reflection and transmission coefficients at dielectric-fractional interface," Progress In Electromagnetics Research, Vol. 125, 543-558, 2012. Google Scholar

21. Balanis, C. A., Advanced Engineering Electromagnetics, Wiley, 1989.

Mr. Muhammad Omar

Dr. Muhammad Junaid Mughal