SpringerBriefs in Applied Sciencesand Technology

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Muhammad Zubair • Muhammad Junaid MughalQaisar Abbas Naqvi

Electromagnetic Fieldsand Waves in FractionalDimensional Space

123

Muhammad ZubairFaculty of Electronic EngineeringGIK Institute of Engineering

Sciences and TechnologyTopiPakistane-mail: zubair_wah@hotmail.com

Muhammad Junaid MughalFaculty of Electronic EngineeringGIK Institute of Engineering

Sciences and TechnologyTopiPakistane-mail: junaid.mughal@gmail.com

Qaisar Abbas NaqviDepartment of ElectronicsQuaid-e-Azam UniversityIslamabadPakistane-mail: nqaisar@yahoo.com

ISSN 2191-530X e-ISSN 2191-5318ISBN 978-3-642-25357-7 e-ISBN 978-3-642-25358-4DOI 10.1007/978-3-642-25358-4Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2011942412

� The Author(s) 2012This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcast-ing, reproduction on microfilm or in any other way, and storage in data banks. Duplication of thispublication or parts thereof is permitted only under the provisions of the German Copyright Law ofSeptember 9, 1965, in its current version, and permission for use must always be obtained fromSpringer. Violations are liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.

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It’s ironic that fractals, many of which wereinvented as examples of pathological behav-ior, turn out to be pathological at all. In factthey are the rule in the universe. Shapes,which are not fractal, are the exception. Ilove Euclidean geometry, but it is quite clearthat it does not give a reasonable presenta-tion of the world. Mountains are not cones,clouds are not spheres, trees are not cylin-ders, neither does lightning travel in astraight line. Almost everything around usis non-Euclidean.

Benoit Mandelbrot, 1924

To my beloved fatherMr. Hafiz Muhammad Makhdoomwhose utmost efforts since my childhoodmake me what I am today

M. Zubair

To my fatherMr. Abdul Ghafoor Mughalfor his love and kindness when he was aliveand his beautiful memories when he is nolonger with us

M. J. Mughal

To my parentsQ. A. Naqvi

Preface

The concept of fractional dimensional space is being effectively used in manyareas of physics to describe the effective parameters of physical systems. Althoughthe space, embedding things, in real world is three dimensional Euclidean space,the material objects are not always moving in three dimensional space. Thedimensionality depends upon the restraint conditions. The phenomenon of elec-tromagnetic wave propagation, radiation and scattering in fractal structures can bedescribed by replacing these confining fractal structures with a D-dimensionalfractional space. Thus, given this simple value of D, the real system can bemodeled in a simple analytical way.

With this view, a theoretical investigation of electromagnetic fields and wavesin fractional dimensional space is provided in this book which is useful to studythe behavior of electromagnetic fields and waves in fractal media. A novel frac-tional space generalization of the differential electromagnetic equations is pro-vided. A new form of vector differential operators is formulated in fractionalspace. Using these modified vector differential operators, the classical Maxwell’selectromagnetic equations have been worked out. The Laplace’s, Poisson’s andHelmholtz’s equations in fractional space are derived by using modified vectordifferential operators. A fractional space generalization of potentials for static andtime-varying fields is presented by solving Laplace’s equation and inhomogeneousvector wave equation, respectively, in fractional space. The phenomenon ofelectromagnetic wave propagation in fractional space is studied in detail byproviding full analytical plane-, cylindrical- and spherical-wave solutions of thevector wave equation in D-dimensional fractional space. An analytical solutionprocedure for radiation problems in fractional space has also been proposed. As anapplication, the fields radiated by a Hertzian dipole in fractional space have beenworked out. For all the investigated cases when integer dimensional space isconsidered, the classical results are recovered. The differential electromagneticequations in fractional space, established in this book, provide a basis forapplication of the concept of fractional space in solving electromagnetic wavepropagation, radiation and scattering problems in fractal media.

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This book has been divided into six chapters. In Chap. 2, a novel generalizationof differential electromagnetic equations in fractional space is provided on thebasis of modified vector differential operators for fractional space. A new form ofvector differential operator Del, written as rD, and its related differential operatorsis formulated in fractional space. Using these modified vector differential opera-tors, the classical Maxwell’s electromagnetic equations have been worked out. TheLaplace’s, Poisson’s and Helmholtz’s equations in fractional space are alsoderived by using modified vector differential operators. Also a new fractionalspace generalization of potentials for static and time-varying fields is presented.Most of the work in later chapters is related to the solution of the establisheddifferential electromagnetic equations in fractional space.

In Chap. 3, a fractional space generalization of potentials for static and time-varying fields is presented by solving Laplace’s equation and inhomogeneousvector wave equation, respectively, in fractional space.

In Chap. 4, the phenomenon of wave propagation in fractional space isinvestigated by solving Helmholtz’s equation in different coordinate systems.General plane wave solutions, in source-free and lossless as well as lossy media, infractional space are presented by solving vector wave equation in D-dimensionalfractional space. An exact solution of cylindrical as well as spherical waveequation, for electromagnetic field in D-dimensional fractional space, is alsopresented. All these investigated solutions of vector wave equation provide a basisfor the application of the concept of fractional space to the wave propagationphenomenon in fractal media. For all investigated cases when integer dimension isconsidered, the classical results were recovered to validate obtained results.

Chapter 5 deals with the solution procedure for radiation problems in fractionalspace.The proposed solution procedure can be used to study the radiation phe-nomenon in any non-integer dimensional fractal media. As an application, thefields radiated by a Hertzian dipole in fractional space have been worked out.Finally, conclusions are drawn in Chap. 6.

In summary, the subject covered in this book is relatively new and emergingarea of research in the field of electromagnetics. The concept of fractionaldimensional space has potential to make a significant impact on future directionsin fractional electromagnetics research. We highly recommend this book tograduate students, researchers, and professionals working in the areas of electro-magnetic-wave propagation, radiation, scattering, diffraction, and other relatedfields of applied mathematics. The topics in this book can also be covered in anygraduate course on ’’Advanced Engineering Electromagnetics’’.

PakistanSeptember 2011

Muhammad ZubairMuhammad Junaid Mughal

Qaisar Abbas Naqvi

x Preface

Acknowledgments

This book is an enlarged form of Authors’ work on fractional dimensional spaceelectromagnetics published in different journals. Some figures from publishedwork have been reproduced with prior permission and are cited with fullacknowledgement to corresponding source.

We would like to sincerely thank the GIK Institute of Engineering Sciences andTechnology, Topi, Pakistan, for providing the necessary facilities to accomplishthis work. We would also take this opportunity to thank all our friends and col-leagues who have helped us in our research work.

Our special thanks goes to our respected Professor Azhar Abbas Rizvi (Ph.D.),Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan. He isthe person who taught us the subject of electromagnetics and nurtured our interestin this field. We feel extremely fortunate to have learnt this subject from him andare sure to say that this work could not have been accomplished without hisguidance.

Finally, we are very thankful to Dr. Christoph Baumann and Mrs. CarmenWolfat Springer-Verlag GmbH for their wonderful help in the preparation and publi-cation of this manuscript.

PakistanSeptember 2011

Muhammad ZubairMuhammad Junaid Mughal

Qaisar Abbas Naqvi

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Fractional Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Axiomatic Basis for Fractional Dimensional Space . . . . . . . . . . 31.3 Differential Geometry of Fractional Dimensional Space . . . . . . . 4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Differential Electromagnetic Equations in Fractional Space . . . . . . 72.1 Fractional Space Generalization of Laplacian Operator . . . . . . . 72.2 Fractional Space Generalization of Del Operator

and Related Differential Operators. . . . . . . . . . . . . . . . . . . . . . 82.2.1 Del Operator in Fractional Space . . . . . . . . . . . . . . . . . 82.2.2 Gradient Operator in Fractional Space . . . . . . . . . . . . . . 102.2.3 Divergence Operator in Fractional Space . . . . . . . . . . . . 102.2.4 Curl Operator in Fractional Space . . . . . . . . . . . . . . . . . 10

2.3 Fractional Space Generalization of DifferentialMaxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Fractional Space Generalization of Potentials for Static Fields,Poisson’s and Laplace’s Equations. . . . . . . . . . . . . . . . . . . . . . 12

2.5 Fractional Space Generalization of Potentialsfor Time-Varying Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Fractional Space Generalization of the Helmholtz’s Equation . . . 152.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Potentials for Static and Time-Varying Fieldsin Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 Electrostatic Potential in Fractional Space . . . . . . . . . . . . . . . . 17

3.1.1 An Exact Solution of the Laplace’s Equationin D-dimensional Fractional Space . . . . . . . . . . . . . . . . 17

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3.1.2 Electrostatic Potential Inside a Rectangular Boxin Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Time-Varying Potentials in Fractional Space. . . . . . . . . . . . . . . 21

3.2.1 Inhomogeneous Vector Potential Wave Equationin D-dimensional Fractional Space . . . . . . . . . . . . . . . . 21

3.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Electromagnetic Wave Propagation in Fractional Space . . . . . . . . 274.1 General Plane Wave Solutions in Fractional Space:

Lossless Medium Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.1 General Plane Wave Solutions in Fractional Space . . . . . 274.1.2 Discussion on Fractional Space Solution . . . . . . . . . . . . 304.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 General Plane Wave Solutions in Fractional Space:Lossy Medium Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2.1 General Plane Wave Solutions in Lossy Medium

in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.2 Discussion on Fractional Space Solution

in Lossy Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.3 Example: Current Sheet as Source of Plane Waves

in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Cylindrical Wave Propagation in Fractional Space . . . . . . . . . . 424.3.1 An Exact Solution of Cylindrical Wave Equation

in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.2 Discussion on Cylindrical Wave Solution

in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Spherical Wave Propagation in Fractional Space . . . . . . . . . . . . 514.4.1 Spherical Wave Equation in D-dimensional

Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4.2 Discussion on Fractional Space Solution . . . . . . . . . . . . 554.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Electromagnetic Radiations from Sources in Fractional Space . . . . 615.1 Solution Procedure for Radiation Problems

in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.1.1 The Vector Potential AD for Electric Current Source J . . 615.1.2 The Vector Potential FD for Magnetic

Current Source M . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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5.1.3 Radiated Electric and Magnetic Fields in Far Zonefor Electric J and Magnetic Current Source M . . . . . . . . 63

5.2 Elementary Hertzian Dipole in Fractional Space . . . . . . . . . . . . 645.2.1 Fields Radiated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2.2 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Contents xv

About the Authors

Mr. Muhammad Zubair did his BS in Electronic Engineering with Gold Medalfrom International Islamic University, Islamabad, Pakistan in 2009. Recently, hehas completed his MS in Electronic Engineering with Highest Distinction fromGIK Institute of Engineering Sciences and Technology, Topi, Pakistan in 2011 andjoined the same institute as Research Associate. Mr. Zubair’s research interests arein the field of Analytical Electromagnetics. He has applied the concept of frac-tional dimensional space in the study of electromagnetic wave propagation,radiation and scattering in fractal media. He is also member of Pakistan Engi-neering Council.

Dr. M. Junaid Mughal did his M.Sc and M.Phil in Electronics from Quaid-e-Azam university, Islamabad in 1993 and 1995, respectively. He did his PhD fromthe University of Birmingham, UK in 2001. He worked as Director of Engineeringin Nuonics Inc., Orlando, Fl, USA form 2001 to 2003. He is presently working asAssociate Professor in the Faculty of Electronic Engineering in GIK Institute.Dr. Mughal’s research interests are primarily in the field of communications andparticularly in RF and Optical Communications. He has worked in antennas, EMscattering, propagation modeling for mobile applications and fiber optics. In thefield of optical communication Dr Mughal is coinventor of high dynamic rangevariable optical attenuators based on Acousto-Optic and MEMS technology, highspeed fiber-optic switches, fiber optic tunable filters and laser beam profilingsystems. Currently he is working in the area of tunable metamaterials, wavepropagation in fractal media and focusing systems embedded in Chiral medium.

Dr. Qaisar Abbas Naqvi completed his M.Sc., M.Phil., and Ph.D., all inElectronics, from Department of Electronics, Quaid-i-Azam University, Islama-bad, Pakistan in 1991, 1993, and 1998 respectively. He joined Department ofElectronics as Assistant Professor in 1998. Uptill now, he has successfullysupervised more than thirty M.Phil and nine PhD students. He is now AssociateProfessor and Chairman of Department of Electronics. He is author of more than

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100 papers in international refereed journals. He is also serving as referee for morethan 10 international journals. His research interests include fractional paradigm inelectromagnetics, bi-isotropic and chiral mediums, high frequency techniques andKobayashi potential method.

xviii About the Authors