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SpringerBriefs in Applied Sciences and Technology

For further volumes: http://www.springer.com/series/8884

Muhammad Zubair Muhammad Junaid Mughal Qaisar Abbas Naqvi •

Electromagnetic Fields and Waves in Fractional Dimensional Space

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Muhammad Zubair Faculty of Electronic Engineering GIK Institute of Engineering Sciences and Technology Topi Pakistan e-mail: [email protected]

Qaisar Abbas Naqvi Department of Electronics Quaid-e-Azam University Islamabad Pakistan e-mail: [email protected]

Muhammad Junaid Mughal Faculty of Electronic Engineering GIK Institute of Engineering Sciences and Technology Topi Pakistan e-mail: [email protected]

ISSN 2191-530X ISBN 978-3-642-25357-7 DOI 10.1007/978-3-642-25358-4

e-ISSN 2191-5318 e-ISBN 978-3-642-25358-4

Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011942412 Ó The Author(s) 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

It’s ironic that fractals, many of which were invented as examples of pathological behavior, turn out to be pathological at all. In fact they are the rule in the universe. Shapes, which are not fractal, are the exception. I love Euclidean geometry, but it is quite clear that it does not give a reasonable presentation of the world. Mountains are not cones, clouds are not spheres, trees are not cylinders, neither does lightning travel in a straight line. Almost everything around us is non-Euclidean. Benoit Mandelbrot, 1924

To my beloved father Mr. Hafiz Muhammad Makhdoom whose utmost efforts since my childhood make me what I am today M. Zubair To my father Mr. Abdul Ghafoor Mughal for his love and kindness when he was alive and his beautiful memories when he is no longer with us M. J. Mughal To my parents Q. A. Naqvi

Preface

The concept of fractional dimensional space is being effectively used in many areas of physics to describe the effective parameters of physical systems. Although the space, embedding things, in real world is three dimensional Euclidean space, the material objects are not always moving in three dimensional space. The dimensionality depends upon the restraint conditions. The phenomenon of electromagnetic wave propagation, radiation and scattering in fractal structures can be described by replacing these confining fractal structures with a D-dimensional fractional space. Thus, given this simple value of D, the real system can be modeled in a simple analytical way. With this view, a theoretical investigation of electromagnetic fields and waves in fractional dimensional space is provided in this book which is useful to study the behavior of electromagnetic fields and waves in fractal media. A novel fractional space generalization of the differential electromagnetic equations is provided. A new form of vector differential operators is formulated in fractional space. Using these modified vector differential operators, the classical Maxwell’s electromagnetic equations have been worked out. The Laplace’s, Poisson’s and Helmholtz’s equations in fractional space are derived by using modified vector differential operators. A fractional space generalization of potentials for static and time-varying fields is presented by solving Laplace’s equation and inhomogeneous vector wave equation, respectively, in fractional space. The phenomenon of electromagnetic wave propagation in fractional space is studied in detail by providing full analytical plane-, cylindrical- and spherical-wave solutions of the vector wave equation in D-dimensional fractional space. An analytical solution procedure for radiation problems in fractional space has also been proposed. As an application, the fields radiated by a Hertzian dipole in fractional space have been worked out. For all the investigated cases when integer dimensional space is considered, the classical results are recovered. The differential electromagnetic equations in fractional space, established in this book, provide a basis for application of the concept of fractional space in solving electromagnetic wave propagation, radiation and scattering problems in fractal media.

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Preface

This book has been divided into six chapters. In Chap. 2, a novel generalization of differential electromagnetic equations in fractional space is provided on the basis of modified vector differential operators for fractional space. A new form of vector differential operator Del, written as rD, and its related differential operators is formulated in fractional space. Using these modified vector differential operators, the classical Maxwell’s electromagnetic equations have been worked out. The Laplace’s, Poisson’s and Helmholtz’s equations in fractional space are also derived by using modified vector differential operators. Also a new fractional space 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 established differential electromagnetic equations in fractional space. In Chap. 3, a fractional space generalization of potentials for static and timevarying fields is presented by solving Laplace’s equation and inhomogeneous vector wave equation, respectively, in fractional space. In Chap. 4, the phenomenon of wave propagation in fractional space is investigated by solving Helmholtz’s equation in different coordinate systems. General plane wave solutions, in source-free and lossless as well as lossy media, in fractional space are presented by solving vector wave equation in D-dimensional fractional space. An exact solution of cylindrical as well as spherical wave equation, for electromagnetic field in D-dimensional fractional space, is also presented. All these investigated solutions of vector wave equation provide a basis for the application of the concept of fractional space to the wave propagation phenomenon in fractal media. For all investigated cases when integer dimension is considered, the classical results were recovered to validate obtained results. Chapter 5 deals with the solution procedure for radiation problems in fractional space.The proposed solution procedure can be used to study the radiation phenomenon in any non-integer dimensional fractal media. As an application, the fields 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 emerging area of research in the field of electromagnetics. The concept of fractional dimensional space has potential to make a significant impact on future directions in fractional electromagnetics research. We highly recommend this book to graduate students, researchers, and professionals working in the areas of electromagnetic-wave propagation, radiation, scattering, diffraction, and other related fields of applied mathematics. The topics in this book can also be covered in any graduate course on ’’Advanced Engineering Electromagnetics’’. Pakistan September 2011

Muhammad Zubair Muhammad Junaid Mughal Qaisar Abbas Naqvi

Acknowledgments

This book is an enlarged form of Authors’ work on fractional dimensional space electromagnetics published in different journals. Some figures from published work have been reproduced with prior permission and are cited with full acknowledgement to corresponding source. We would like to sincerely thank the GIK Institute of Engineering Sciences and Technology, Topi, Pakistan, for providing the necessary facilities to accomplish this work. We would also take this opportunity to thank all our friends and colleagues 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 is the person who taught us the subject of electromagnetics and nurtured our interest in this field. We feel extremely fortunate to have learnt this subject from him and are sure to say that this work could not have been accomplished without his guidance. Finally, we are very thankful to Dr. Christoph Baumann and Mrs. CarmenWolf at Springer-Verlag GmbH for their wonderful help in the preparation and publication of this manuscript. Pakistan September 2011

Muhammad Zubair Muhammad Junaid Mughal Qaisar Abbas Naqvi

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fractional Dimensional Space . . . . . . . . . . . . . . . . . . . 1.2 Axiomatic Basis for Fractional Dimensional Space . . . . 1.3 Differential Geometry of Fractional Dimensional Space . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 3 4 5

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Differential Electromagnetic Equations in Fractional Space . . . . . 2.1 Fractional Space Generalization of Laplacian Operator . . . . . . 2.2 Fractional Space Generalization of Del Operator and Related Differential Operators. . . . . . . . . . . . . . . . . . . . . 2.2.1 Del Operator in Fractional Space . . . . . . . . . . . . . . . . 2.2.2 Gradient Operator in Fractional Space . . . . . . . . . . . . . 2.2.3 Divergence Operator in Fractional Space . . . . . . . . . . . 2.2.4 Curl Operator in Fractional Space . . . . . . . . . . . . . . . . 2.3 Fractional Space Generalization of Differential Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fractional Space Generalization of Potentials for Static Fields, Poisson’s and Laplace’s Equations. . . . . . . . . . . . . . . . . . . . . 2.5 Fractional Space Generalization of Potentials for Time-Varying Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Fractional Space Generalization of the Helmholtz’s Equation . . 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Potentials for Static and Time-Varying Fields in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Electrostatic Potential in Fractional Space . . . . . . . . . . . . . . . . 3.1.1 An Exact Solution of the Laplace’s Equation in D-dimensional Fractional Space . . . . . . . . . . . . . . . .

17 17 17

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3.1.2

Electrostatic Potential Inside a Rectangular Box in Fractional Space . . . . . . . . . . . . . . . . . . . . . 3.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Time-Varying Potentials in Fractional Space. . . . . . . . . 3.2.1 Inhomogeneous Vector Potential Wave Equation in D-dimensional Fractional Space . . . . . . . . . . 3.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5

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21 25 25

Electromagnetic Wave Propagation in Fractional Space . . . . 4.1 General Plane Wave Solutions in Fractional Space: Lossless Medium Case . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 General Plane Wave Solutions in Fractional Space . 4.1.2 Discussion on Fractional Space Solution . . . . . . . . 4.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 General Plane Wave Solutions in Fractional Space: Lossy Medium Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 General Plane Wave Solutions in Lossy Medium in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Discussion on Fractional Space Solution in Lossy Medium . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Example: Current Sheet as Source of Plane Waves in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Cylindrical Wave Propagation in Fractional Space . . . . . . 4.3.1 An Exact Solution of Cylindrical Wave Equation in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Discussion on Cylindrical Wave Solution in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Spherical Wave Propagation in Fractional Space . . . . . . . . 4.4.1 Spherical Wave Equation in D-dimensional Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Discussion on Fractional Space Solution . . . . . . . . 4.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Electromagnetic Radiations from Sources in Fractional Space . . 5.1 Solution Procedure for Radiation Problems in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Vector Potential AD for Electric Current Source J 5.1.2 The Vector Potential FD for Magnetic Current Source M . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.1.3

Radiated Electric and Magnetic Fields in Far Zone for Electric J and Magnetic Current Source M . . . . 5.2 Elementary Hertzian Dipole in Fractional Space . . . . . . . . 5.2.1 Fields Radiated . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

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63 64 64 66 67 67

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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About the Authors

Mr. Muhammad Zubair did his BS in Electronic Engineering with Gold Medal from International Islamic University, Islamabad, Pakistan in 2009. Recently, he has completed his MS in Electronic Engineering with Highest Distinction from GIK Institute of Engineering Sciences and Technology, Topi, Pakistan in 2011 and joined the same institute as Research Associate. Mr. Zubair’s research interests are in the field of Analytical Electromagnetics. He has applied the concept of fractional dimensional space in the study of electromagnetic wave propagation, radiation and scattering in fractal media. He is also member of Pakistan Engineering Council. Dr. M. Junaid Mughal did his M.Sc and M.Phil in Electronics from Quaid-eAzam university, Islamabad in 1993 and 1995, respectively. He did his PhD from the University of Birmingham, UK in 2001. He worked as Director of Engineering in Nuonics Inc., Orlando, Fl, USA form 2001 to 2003. He is presently working as Associate Professor in the Faculty of Electronic Engineering in GIK Institute. Dr. Mughal’s research interests are primarily in the field of communications and particularly in RF and Optical Communications. He has worked in antennas, EM scattering, propagation modeling for mobile applications and fiber optics. In the field of optical communication Dr Mughal is coinventor of high dynamic range variable optical attenuators based on Acousto-Optic and MEMS technology, high speed fiber-optic switches, fiber optic tunable filters and laser beam profiling systems. Currently he is working in the area of tunable metamaterials, wave propagation in fractal media and focusing systems embedded in Chiral medium. Dr. Qaisar Abbas Naqvi completed his M.Sc., M.Phil., and Ph.D., all in Electronics, from Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan in 1991, 1993, and 1998 respectively. He joined Department of Electronics as Assistant Professor in 1998. Uptill now, he has successfully supervised more than thirty M.Phil and nine PhD students. He is now Associate Professor 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 more than 10 international journals. His research interests include fractional paradigm in electromagnetics, bi-isotropic and chiral mediums, high frequency techniques and Kobayashi potential method.

Chapter 1

Introduction

This book is a theoretical investigation of electromagnetic fields and waves in the fractional dimensional space. The motivation for this study, besides its theoretical importance, is provided by its applicability to the problems of electromagnetic wave modeling in complex fractal media. One of the important advantages of fractals is their capability to model objects of complicated structures. This is because of an important property of fractals that their structure is characterized by a small number of parameters. One of those parameters is the fractional dimension which tells how the fractal fills the Euclidean space in which it lies. Since, a medium composed of such fractal objects can be considered as non-integer dimensional fractal media, the analytical results of this work provide the necessary tools for analyzing the behavior of electromagnetic fields and waves in it.

1.1 Fractional Dimensional Space Every one of us has learnt that the lines and curves are one-dimensional, planes and surfaces are two-dimensional, solids and volumes are three dimensional, and so on. In a formal way, we say that a set is n-dimensional if we have n independent variables to describe a neighborhood of any point. Such a notion of dimension is called the “topological dimension” of a set. Also we observe that if we take the union of infinite many sets of n dimension, the overall dimension of new set can grow to n+1 e.g., the union of infinite number of (one-dimensional) lines give rise to a (two-dimensional) plane. Now, we think about the dimension in another way. We may break a line segment into 4 self-similar intervals, each with the same length, and each of which can be magnified by a factor of 4 to yield the original segment. We can also break a line segment into 7 self-similar pieces, each with magnification factor 7 can yield the original segment, or 20 self-similar pieces with magnification factor 20 to yield the original segment. In general, we can break a line segment into N self-similar pieces, each with magnification factor N to yield the original segment. If, we decompose a

M. Zubair et al., Electromagnetic Fields and Waves in Fractional Dimensional Space, SpringerBriefs in Applied Sciences and Technology, DOI: 10.1007/978-3-642-25358-4_1, © The Author(s) 2012

1

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1 Introduction

square into 4 self-similar sub-squares, and the magnification factor here will be 2 to yield in the original square. Alternatively, we can break the square into 9 self-similar pieces with magnification factor 3, or 25 self-similar pieces with magnification factor 5. Clearly, the square may be broken into N 2 self-similar copies of itself, each of which must be magnified by a factor of N to yield the original figure. Finally, we can decompose a cube into N 3 self-similar pieces, each of which has magnification factor N. Following above discussion we can say that the dimension is simply the exponent of the number of self-similar pieces with magnification factor N into which the figure may be broken. So what is the dimension of the Sierpinski triangle? How do we find the exponent in this case? For this, we need logarithms. Note that, for the square, we have N 2 self-similar pieces, each with magnification factor N. So we can write dimension =

log N 2 log(number of self similar pieces) = =2 log(magnification factor) log N

Similarly, the dimension of a cube is: dimension =

log N 3 log(number of self similar pieces) = =3 log(magnification factor) log N

Thus, we take as the definition of the fractal dimension of a self-similar object: dimension =

log(number of self similar pieces) log(magnification factor)

As the Sierpinski triangle consists of 3 self-similar pieces, each with magnification factor 2, So the fractal dimension is dimension =

log(number of self similar pieces) log 3 = ≈ 1.58 log(magnification factor) log 2

In general, the Sierpinski triangle breaks into 3N self-similar pieces with magnification factors 2N , so we again have fractal dimension =

log 3N log(number of self similar pieces) = ≈ 1.58 log(magnification factor) log 2N

This estimates that the Sierpinski triangle sits somewhere in between lines and planes. Similarly, many fractal structures are known in literature that possess a fractal dimension. Roughly speaking, we state that the space embedding such fractal curves or surfaces is known as “fractional dimensional space”. There has been much interest to study different physical phenomenon in fractional dimensional space [1–28] during the last few decades. The concept of fractional space is used to replace the real anisotropic confining structure with an isotropic fractional

1.1 Fractional Dimensional Space

3

space, where the measurement of this confinement is given by fractional dimension [6, 7]. It is also important to mention that the experimental measurement of the dimension of real world is 3 ± 10−6 , not exactly 3 [6, 9]. Among several methods, a methodology to describe the fractional dimension is fractional calculus [29], which is also used by different authors [30–37] in studying various electromagnetic problems. Axiomatic basis for spaces with fractional dimension have been provided by Stillinger [6], along with a fractional space generalization of Laplacian operator and a solution of Schrödinger’s wave equation in fractional dimensional space. For 2-spatial coordinate space, the Stillinger’s formalism shows that it is possible to distribute the D dimensions between them. Palmer and Stavrinou [8] generalized the results of Stillinger to n orthogonal coordinates. Equations of motion in a non-integer dimensional space have also been formulated in [8]. The formalism investigated in [8] allows to describe an anisotropic confinement of fractional space, i.e., if we have a system that is confined as 1.8 dimensional, then it could be described as 1 + 0.8 dimensional in two coordinates and as 1 + 0.2 + 0.6 dimensional in three coordinates. Recently, Muslih [18] provided a dimensional regularization technique in order to convert any integral of a function from fractional dimensional space to a regular dimensional space along with a description of differential geometry of fractional dimensional space. The generalization of electromagnetic theory in fractional space is of much importance to study the phenomenon of wave propagation, radiation and scattering in an anisotropic fractal media. Fractal models of media are becoming popular due to relatively small number of parameters that define a medium of greater complexity and rich structure [15]. In general, the fractal media cannot be considered as continuous media, because some of points and domains are not filled by the medium particles. These unfilled domains are called porous. The fractal media can be treated as continuous media for the scales much larger than average pore size. In order to describe the fractal media, the continuous medium model for fractal media reported in [16], suggests to use the space with fractional dimension. An introductory work on fractional multipoles and electromagnetic field in fractional space is reported in [17–20]. It is worthwhile to mention that clouds, turbulence in fluids, rough surfaces, snow, etc., can be described as fractional dimensional. The study of wave propagation, radiation and scattering phenomenon in such media is important in practical applications, such as communications, remote sensing, navigation and even bioengineering [20].

1.2 Axiomatic Basis for Fractional Dimensional Space This work is based on the Stillinger’s [6] axiomatic basis for spaces with non-integer dimension. Here, we briefly describe these axioms. In [6], four topological axioms are proposed which generate a space with non-integer dimension D. Let SD denote the fractional space which contains points x, y, . . . and has topological structure specified by the following axioms: Axiom 1 SD is a metric space.

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1 Introduction

Axiom 2 SD is dense in itself. Axiom 3 SD is metrically unbounded. Axiom 4 For any two points y, z ∈ SD , and any ∈ >0, there exists an x ∈ SD such that: (a) (b)

r(x, y) + r(x, z) = r(y, z) |r(x, y) − r(x, z)| < εr(y, z)

The full implication of Axiom 4 is that any two points in SD are connected by a continuous line embedded in that space so SD is connected. So any convex or star domain in SD will be contractible. Based on these axioms Stillinger [6] as well as recently Muslih [18], provided a dimensional regularization technique in order to convert any integral of a function from fractional dimensional space to a regular dimensional space. The fractional space generalization of the Laplacian operator, provided in later chapters, is based on the same dimensional regularization technique according to which a fractional space is related to fractional integrals and derivatives.

1.3 Differential Geometry of Fractional Dimensional Space The question of differential geometry of fractional spaces is related to the dimensional regularization technique. A sufficient description of differential geometry of fractional space along with dimensional regularization technique is already provided in [18] which is briefly described here: Let us say that N coordinates x1 , x2 , . . . , xN are needed to locate a point in a space. In the case where a space is filled with regular geometric objects, and the curves and the surfaces are smooth, it is common to call this number N as the dimension of the space. Thus, a straight line, a plane surface, and a cube are of dimensions 1, 2, and 3, respectively. This is also true if these spaces have curvatures. For example, motions along the circumference of a circle and on the surface of a sphere can be considered as motions in one- and two-dimensional spaces even though our true motions may be in a three-dimensional space. In such cases, infinitesimal line, area, and volume elements in the Cartesian coordinates are defined as dx1 , dx1 dx2 , and dx1 dx2 dx3 , respectively, and even in the case of a space with curvature, the distance between two points sufficiently closed to each other is given by a quadratic expression. However, this is not the situation in the case of fractal lines, surfaces, volume and hypervolumes. Thus, in these cases, there is a clear distinction between the number of coordinates used to locate a point and the dimension of the space. The dimensions of the fractional space can be defined in various ways. In [18], a scaling method d α x = f (α)|x|α−1 dx is used to relate the differential fractional line element d α x with dx, where 0 < α ≤ 1 is the fractional dimension of the line and α/2 f (α) = Γπ(α/2) is a function of α. A space filled with such lines is called fractional space. Thus, if a point in a space is located using N points x1 , x2 , . . . , xN , then the

1.3 Differential Geometry of Fractional Dimensional Space

5

scaling between d αi xi and dxi is defined as d αi xi =

π αi /2 |x|αi −1 dxi , i = 1, 2, . . . , N Γ (αi /2)

This space is called fractional dimensional space or simply a fractional space, and D = α1 + α2 +, . . . , +αN is the dimension of the fractional space. Following the above discussion, any volume element in D-dimensional fractional space can be defined as: d(V )D = d α1 x1 , d α2 x1 , . . . , d αN xN where, d αN xN is the differential element corresponding to Nth coordinate. In later chapters, we have established the differential electromagnetic equations in fractional dimension space to provide a basis for application of the concept of fractional dimensional space in solving electromagnetic wave propagation, radiation and scattering problems in fractal media.

References 1. M. Zubair, M.J. Mughal, Q.A. Naqvi, The wave equation and general plane wave solutions in fractional space. Prog. Electromagnet. Res. Lett. 19, 137–146 (2010) 2. M. Zubair, M.J. Mughal, Q.A. Naqvi, On electromagnetic wave propagation in fractional space. Non-linear Anal. B: Real World App. 12(5), 2844–2850 (2011) 3. M. Zubair, M.J. Mughal, Q.A. Naqvi, A.A. Rizvi, Differential electromagnetic equations in fractional space. Prog. Electromagnet. Res. 114, 255–269 (2011) 4. M. Zubair, M.J. Mughal, Q.A. Naqvi, An exact solution of cylindrical wave equation for electromagnetic field in fractional dimensional space. Prog. Electromagnet. Res. 114, 443–455 (2011) 5. M. Zubair, M.J. Mughal, Q.A. Naqvi, An exact solution of spherical wave in D-dimensional fractional space. J. Electromagn. Waves App. 25, 1481–1491 (2011) 6. F.H. Stillinger, Axiomatic basis for spaces with noninteger dimension. J. Math. Phys. 18(6), 1224–1234 (1977) 7. X. He, Anisotropy and isotropy: a model of fraction-dimensional space. PSolid State Commun. 75, 111–114 (1990) 8. C. Palmer, P.N. Stavrinou, Equations of motion in a noninteger-dimension space. J. Phys. A 37, 6987–7003 (2004) 9. K.G. Willson, Quantum field-theory, models in less than 4 dimensions. Phys. Rev. D 7(10), 2911–2926 (1973) 10. B. Mandelbrot, The Fractal Geometry of Nature. (W.H. Freeman, New York, 1983) 11. C.G. Bollini, J.J. Giambiagi, Dimensional renormalization: The number of dimensions as a regularizing parameter. Nuovo Cimento B 12, 20–26 (1972) 12. J.F. Ashmore, On renormalization and complex space-time dimensions. Commun. Math. Phys. 29, 177–187 (1973) 13. O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 271(1), 368–379 (2002) 14. D. Baleanu, S. Muslih, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scripta 72(23), 119–121 (2005)

6

1 Introduction

15. V.E. Tarasov, Electromagnetic fields on fractals. Modern Phys. Lett. A 21(20), 1587–1600 (2006) 16. V.E. Tarasov, Continuous medium model for fractal media. Phys. Lett. A 336(2–3), 167–174 (2005) 17. S. Muslih, D. Baleanu, Fractional multipoles in fractional space. Nonlinear Anal: Real World App. 8, 198–203 (2007) 18. S.I. Muslih, O.P. Agrawal, A scaling method and its applications to problems in fractional dimensional space. J. Math. Phys. 50(12):123501–123511 (2009) 19. D. Baleanu, A.K. Golmankhaneh, A.K. Golmankhaneh, On electromagnetic field in fractional space. Nonlinear Anal: Real World App. 11(1):288–292 (2010) 20. Z. Wang, B. Lu, The scattering of electromagnetic waves in fractal media. Waves Random Complex Media 4(1), 97–103 (1994) 21. C.M. Bender, K.A. Milton, Scalar casimir effect for a D-dimensional sphere. Phys. Rev. D 50, 6547–6555 (1994) 22. S. Muslih, D. Baleanu, Mandelbrot scaling and parametrization invariant theories. Romanian Rep. Phys. 62(4), 689–696 (2010) 23. S. Muslih, M. Saddallah, D. Baleanu, E. Rabe, Lagrangian formulation of maxwell’s field in fractional D dimensional space-time. Romanian Rep. Phys. 55(7–8), 659–663 (2010) 24. S. Muslih, O.P. Agrawal, Riesz fractional derivatives and fractional dimensional space. Int. J. Theor. Phys. 49(2):270–275 (2010) 25. S. Muslih, Solutions of a particle with fractional [delta]-potential in a fractional dimensional space. Int. J. Theor. Phys. 49(9), 2095–2104 (2010) 26. E. Rajeh, S.I. Muslih, B. Dumitru, E. Rabei, On fractional Schrodinger equation in [alpha]dimensional fractional space. Nonlinear Anal.: Real World App. 10(3):1299–1304 (2009) 27. M. Sadallah, S.I. Muslih, Solution of the equations of motion for einsteins field in fractional D dimensional space-time. Int. J. Theor. Phys. 48(12):3312–3318 (2009) 28. S. Muslih, D. Baleanu, E.M. Rabe, Solutions of massless conformal scalar field in an n-dimensional einstein space. Acta Phys. Pol. Ser. B 39(4):887–892 (2008) 29. K.B. Oldham, J. Spanier, The Fractional Calculus. (Academic Press, New York, 1974) 30. A. Hussain, Q.A. Naqvi, Fractional rectangular impedance waveguide. Prog. Electromagnet. Res. 96, 101–116 (2009) 31. Q.A. Naqvi, Planar slab of chiral nihility metamaterial backed by fractional dual/PEMC interface. Prog. Electromagnet. Res. 85, 381–391 (2008) 32. Q.A. Naqvi, Fractional dual interface in chiral nihility medium. Prog. Electromagnet. Res. Lett. 8, 135–142 (2009) 33. Q.A. Naqvi, Fractional dual solutions in grounded chiral nihility slab and their effect on outside fields. J. Electromagn. Waves App. 23(5–6), 773–784(12) (2009) 34. A. Naqvi, S. Ahmed, Q.A. Naqvi, Perfect electromagnetic conductor and fractional dual interface placed in a chiral nihility medium. J. Electromagn. Waves App. 24(14–15), 1991–1999(9) (2010) 35. A. Naqvi, A. Hussain, Q.A. Naqvi, Waves in fractional dual planar waveguides containing chiral nihility metamaterial. J. Electromagn. Waves App. 24(11–12), 1575–1586(12) (2010) 36. E.I. Veliev, M.V. Ivakhnychenko, T.M. Ahmedov, Fractional boundary conditions in plane waves diffraction on a strip. Prog. Electromagn. Res. 79, 443–462 (2008) 37. S.A. Naqvi, M. Faryad, Q.A. Naqvi, M. Abbas, Fractional duality in homogeneous bi-isotropic medium. Prog. Electromagn. Res. 78, 159–172 (2008)

Chapter 2

Differential Electromagnetic Equations in Fractional Space

In this chapter a novel generalization of differential electromagnetic equations in fractional space is provided. Firstly, basic vector differential operators are generalized in fractional space and then using these fractional operators Maxwell’s, Laplace’s, Poisson’s and Helmholtz’s equations have been worked out in fractional space. The differential electromagnetic equations in fractional space, established in this chapter, provide a basis for application of the concept of fractional space in practical electromagnetic wave propagation and scattering problems in fractal media. In Sect. 2.1 a review of already existing study to construct a generalized Laplacian operator using integration in D-dimensional fractional space is briefly described. In Sect. 2.2, fractional space generalization of the Del operator, written as ∇ D , and its related differential operators ( i.e., gradient, divergence and curl) in vector calculus is obtained. In Sect. 2.3, a novel fractional space generalization of differential Maxwell’s equations is presented. In Sect. 2.4, fractional space generalization of the Laplace and Poisson’s equations is established in addition to fractional space generalization of potentials for static field. In Sect. 2.5, potentials for time varying fields in fractional space are derived. In Sect. 2.6, the Helmholtz’s equation in fractional space is established. Finally, this chapter is summarized in Sect. 2.7.

2.1 Fractional Space Generalization of Laplacian Operator In [1] a formalism is provided for integration on D-dimensional fractional space. According to this formalism, the integration of radially symmetric function f (r ) in a D-dimensional fractional space is given by [1]: ∞ dx0 f (r (x0 , x1 )) = dr W (r ) f (r ) (2.1) 0

where r (x0 , x1 ) is the distance between two points x0 and x1 , and weight W (r ) given by W (r ) = σ (D)r D−1 M. Zubair et al., Electromagnetic Fields and Waves in Fractional Dimensional Space, SpringerBriefs in Applied Sciences and Technology, DOI: 10.1007/978-3-642-25358-4_2, © The Author(s) 2012

(2.2) 7

8

2 Differential Electromagnetic Equations in Fractional Space

with σ (D) =

2π D/2 Γ (D/2)

(2.3)

From this a single variable Laplacian operator is derived in D-dimensional fractional space as: 2 ∂ D−1 ∂ 2 ∇ D f (r ) = f (r ), 0 < D ≤ 1 (2.4) + ∂r 2 r ∂r In Eq. 2.4 and throughout the discussion, the subscript D is used to emphasize the dimension of space in which this operator is defined. An extension of formalism in Eq. 2.1 to two variable integration yields an expression for a two-coordinate Laplacian operator in fractional space. 2 = ∇D

∂2 ∂2 D−2 ∂ + + , 0 1, ignoring terms involving second or higher degree of x in denominator, leads to the following form: |∇ D | =

∂ 1 D−1 + ∂x 2 x

(2.10)

From (2.8) and (2.10), Del operator in single variable x with fractional dimension D is given by: ∂ 1 D−1 ∇D = + xˆ (2.11) ∂x 2 x Extending above procedure to three variable case for | x |, | y |, | z | 1 we get Del operator ∇ D in fractional space as follows: ∂ 1 α1 − 1 1 α2 − 1 ∂ + + xˆ + yˆ ∇D = ∂x 2 x ∂y 2 y ∂ 1 α3 − 1 + + zˆ (2.12) ∂z 2 z where, parameters (0 < α1 ≤ 1, 0 < α2 ≤ 1 and 0 < α3 ≤ 1) are used to describe the measure distribution of space where each one is acting independently on a single coordinate and the total dimension of the system is D = α1 + α2 + α3 . It is important to mention that Eq. 2.12 and all differential operators presented in later sections are valid in far-field region only i.e (|x|, |y|, |z| 1) because of the first order approximation given by (2.10). Clearly, for three dimensional space (D = 3), if we set α1 = α2 = α3 = 1 in (2.12), the fractional Del operator ∇ D reduces to the classical Del operator ∇ [3] in Euclidean space.

10

2 Differential Electromagnetic Equations in Fractional Space

2.2.2 Gradient Operator in Fractional Space The gradient of a scalar field ψ in fractional space is a vector that represents both the magnitude and the direction of maximum space rate of increase of ψ in fractional space. Using (2.12) the modified form of the gradient of scalar field ψ, written as grad D ψ, in far-field region in the fractional space is given as: ∂ψ ∂ψ 1 (α1 − 1)ψ 1 (α2 − 1)ψ grad D ψ = ∇ D ψ = + + xˆ + yˆ ∂x 2 x ∂y 2 y ∂ψ 1 (α3 − 1)ψ + + zˆ (2.13) ∂z 2 z

2.2.3 Divergence Operator in Fractional Space From (2.12) a generalized form of divergence of a vector F = Fx xˆ + Fy yˆ + Fz zˆ at point P(x0 , y0 , z 0 ) in far-field region in the fractional space is written as div D F and is given by div D F = ∇ D · F =

∂ Fy ∂ Fx 1 (α1 − 1)Fx 1 (α2 − 1)Fy + + + ∂x 2 x ∂y 2 y 1 (α3 − 1)Fz ∂ Fz + + ∂z 2 z

(2.14)

2.2.4 Curl Operator in Fractional Space The modified form of curl of a vector F = Fx xˆ + Fy yˆ + Fz zˆ at point P(x 0 , y0 , z 0 ) in far-field region in the fractional space is written as curl D F and using (2.12) it is given by curl D F = ∇ D × F ∂ Fz + = ∂y ∂ Fx + ∂z ∂ Fy + ∂x

∂ Fy 1 (α2 − 1)Fz + − 2 y ∂z ∂ Fz 1 (α3 − 1)Fx + − 2 z ∂x ∂ Fx 1 (α1 − 1)Fy + − 2 x ∂y

1 (α3 − 1)Fy xˆ 2 z 1 (α1 − 1)Fz + yˆ 2 x 1 (α2 − 1)Fx + zˆ 2 y

(2.15)

or curl D F = ∇ D

yˆ xˆ zˆ ∂ 1 α1 − 1 ∂ 1 α2 − 1 ∂ 1 α3 − 1 + + + × F= (2.16) ∂x 2 x ∂y 2 y ∂z 2 z Fx Fy Fz

2.3 Fractional Space Generalization of Differential Maxwell’s Equations

11

2.3 Fractional Space Generalization of Differential Maxwell’s Equations The Maxwell’s equations are the fundamental equations describing the behavior of electric and magnetic fields. In classical electromagnetic theory following quantities are dealt with: E = electric field intensity (V /m) B = magnetic field intensity (A/m) D = electric flux density (C/m 2 ) B = magnetic flux density (W/m 2 ) J = electric current density (A/m 2 ) ρv = electric charge density (C/m 3 ) with B = μH and D = εE, where μ and ε are permeability and permittivity of the medium, respectively. All of these quantities are functions of space variables x, y, z and tim t. The basic classical Maxwell’s equations in differential form in Euclidean space are [4]: ∇ · D = ρv

(2.17)

∇ ·B=0

(2.18)

∇ × E=−

∂B ∂t

∇ × H=J+

(2.19)

∂D ∂t

(2.20)

Also the continuity equation ∇ ·J=−

∂ρv ∂t

(2.21)

is implicit in Maxwell’s equations. Now we wish to have a generalized form of Maxwell’s equations in D-dimensional fractional space. From the results of Sect. 3, we are now able to write differential form of Maxwell’s equations in far-field region in the fractional space as follows: div D D = ρv

(2.22)

div D B = 0

(2.23)

curl D E = −

∂B ∂t

curl D H = J +

∂D ∂t

(2.24) (2.25)

12

2 Differential Electromagnetic Equations in Fractional Space

and the continuity equation in fractional space as: div D J = −

∂ρv ∂t

(2.26)

where, div D and curl D are defined in Eqs. 2.14–2.16. Equations 2.22–2.25 provide generalization of classical Maxwell’s equations form integer dimensional Euclidean space to a non-integer dimensional fractional space. For D = 3, these fractional equations can be reduced to classical Maxwell’s equations in Euclidean space. In phasor form, assuming a time factor e jωt , Maxwell’s equations in fractional space are given by replacing ∂t∂ with jω [4] as below: div D Ds = ρvs

(2.27)

div D Bs = 0

(2.28)

curl D Es = − jωBs

(2.29)

curl D Hs = Js + jωDs

(2.30)

and the phasor form of continuity equation in fractional space as: div D Js = − jωρvs

(2.31)

where, Ds , Bs , Es , Hs , Js , ρvs represent the phasor form of instantaneous quantities D, B, E, H, J and ρv , respectively.

2.4 Fractional Space Generalization of Potentials for Static Fields, Poisson’s and Laplace’s Equations From Maxwell’s equations in previous section, it is shown that the behavior of electrostatic field in fractional space can be described by two differential equations: div D E =

ρv ε0

curl D E = 0

(2.32) (2.33)

where, ε0 is permittivity of free space. Equation 2.33 being equivalent to the statement that E is the gradient of a scalar function, the scalar potential for electric field ψ. Because curl D (−grad D ψ) = 0

(2.34)

2.4

Fractional Space Generalization

13

so, E = −grad D ψ

(2.35)

A detailed proof of Eq. 2.34 is provided in Appendix A. Equations 2.32 and 2.35 can be combined into one partial differential equation for the single function ψ(x, y, z) as follows: div D grad D ψ =

ρv ε0

(2.36)

2 ψ, so finally we get As div D grad D ψ = ∇ D 2 ∇D ψ=

ρv ε0

(2.37)

2 is scalar Laplacian operator in fractional space given by (2.6). Equation where ∇ D 2.37 is called Poisson’s equation in fractional space. In regions of space that lack a charge density, the scalar potential ψ satisfies the Laplace’s equation given by: 2 ∇D ψ =0

(2.38)

Equations 2.37–2.38 are important in solving practical electrostatic problems in fractional space. From Maxwell’s equations in last section, it is shown that the behavior of magnetostatic field in fractional space can be described by two differential equations: div D B = 0

(2.39)

curl D H = J

(2.40)

From Eq. 2.40 we say that in problems concerned with finding the magnetic fields in a current free region, the curl D of magnetic field H is zero. Any vector with zero curl D may be represented as the grad D of a scalar (see e.g., Eq. 2.34). Thus, the magnetic field for points in such regions can be expressed as H = −grad D ψm

(2.41)

where, ψm (in amperes) is the magnetic scalar potential and the minus sign is taken to complete the analogy with electrostatic field in (2.35). From (2.39), the divergence of B is zero everywhere, so using (2.39) and (2.41) div D (μgrad D ψm ) = 0

(2.42)

Thus for a homogenous medium in fractional space the magnetic scalar potential ψm satisfies the Laplace equation:

14

2 Differential Electromagnetic Equations in Fractional Space 2 ∇D ψm = 0

(2.43)

From (2.39) we know that for magnetostatic field div D B = 0. Also we know that div D curl D A = 0

(2.44)

In order to satisfy (2.39) and (2.44) simultaneously, we can define vector magnetic potential A (in webers/meter) such that B = curl D A

(2.45)

Now if we substitute (2.45) into (2.40) we get curl D curl D A = μJ

(2.46)

This may be considered as differential equation relating A to the current density J. Using vector identity 2 A curl D curl D A = grad D (div D A) − ∇ D

(2.47)

div D A = 0

(2.48)

2 ∇D A = −μJ

(2.49)

with

in (2.46) we get

This is a vector equivalent of Poisson’s equation in (2.37). It includes three component scalar equations which are exactly of the poisson form.

2.5 Fractional Space Generalization of Potentials for Time-Varying Fields A we have seen, in Maxwell’s equations fields are related to each other and sources as well. But sometimes it is helpful to introduce some intermediate functions, known as potential functions, which are directly related to sources and from which we can drive fields [4]. Such function are found useful for static fields as well (see e.g., Eqs. 2.35, 2.41, 2.45). From (2.45) we have B = curl D A. This relation may now be substituted into Maxwell’s equation (2.24) to get ∂A =0 (2.50) curl D E + ∂t

2.5

Fractional Space Generalization of Potentials for Static Fields

15

Equation 2.50 states that curl D of a certain quantity is zero. But this condition allows a vector to be derived as a grad D of a scalar ψ. ∂A = −grad D ψ ∂t ∂A E = −grad D ψ − ∂t

E+

(2.51) (2.52)

Equation 2.45 and 2.52 are the valid relationships between fields and potential functions A and ψ. We substitute (2.52) into (2.22), to obtain 2 −∇ D ψ−

ρv ∂(div D A) = ∂t ε

(2.53)

Then by substituting (2.45) and (2.52) into (2.53), we get

∂ψ ∂ 2A curl D curl D A = μJ + με −grad D − 2 ∂t ∂t

(2.54)

Using the vector identity (2.45) and choosing div D A = −με

∂ψ ∂t

(2.55)

Equation 2.53 and 2.54 can be reduced to 2 ψ − με ∇D

∂ 2ψ ρv =− 2 ∂t ε

(2.56)

2 ∇D A − με

∂ 2A = −μJ ∂t 2

(2.57)

Thus the potential functions A and ψ, defined in terms of sources J and ρv by the Eqs. 2.56 and 2.57 in fractional space, may be used to drive electric and magnetic fields using (2.45) and (2.52).

2.6 Fractional Space Generalization of the Helmholtz’s Equation From Eqs. 2.24 and 2.25, using B = μH and D = εE, we finally obtain ∂H ∂t ∂E curl D H = J + ε ∂t curl D E = −μ

Taking curl D of Eq. 2.58 on both sides and using (2.59) gives

(2.58) (2.59)

16

2 Differential Electromagnetic Equations in Fractional Space

curl D curl D E = −μ

∂E ∂ J+ε ∂t ∂t

(2.60)

This result can be simplified using (2.47) and (2.26) in (2.60) as : ∂ 2E 1 ∂J grad D ρv + μ + με 2 ε ∂t ∂t For source-free region (ρv = 0, J = 0) (2.61) becomes 2 ∇D E=

(2.61)

∂ 2E =0 (2.62) ∂t 2 Equation 2.62 is the Helmholtz’s equation, or wave equation, for E in fractional space. An identical equation for H in fractional space can also be derived in the same manner: 2 ∇D E − με

2 H − με ∇D

∂ 2H =0 ∂t 2

(2.63)

2.7 Summary A novel fractional space generalization of the differential electromagnetic equations, that is helpful in studying the behavior of electric and magnetic fields in fractal media, is provided. A new form of vector differential operator Del, written as ∇ D , and its related differential operators is formulated in fractional space. Using these modified vector differential operators, the classical Maxwell’s electromagnetic equations have been worked out. The Laplace’s, Poisson’s and Helmholtz’s equations in fractional space are derived by using modified vector differential operators. Also a new fractional space generalization of potentials for static and time-varying fields is presented. For all investigated cases, when integer dimensional space is considered, the classical results can be recovered. The provided fractional space generalization of differential electromagnetic equations is valid in far-field region only. The differential electromagnetic equations in fractional space, established in this work, provide a basis for application of the concept of fractional space in practical electromagnetic wave propagation and scattering phenomenon in far-field region in any fractal media.

References 1. F.H. Stillinger, Axiomatic basis for spaces with noninteger dimension. J. Math. Phys. 18(6), 1224–1234 (1977) 2. C. Palmer, P.N. Stavrinou, Equations of motion in a noninteger-dimension space. J Phys A 37, 6987–7003 (2004) 3. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables. (Department of Commerce, U.S., 1972) 4. C.A. Balanis, Advanced Engineering Electromagnetics. (Wiley, New York, 1989)

Chapter 3

Potentials for Static and Time-Varying Fields in Fractional Space

In this chapter, a fractional space generalization of potentials for static and timevarying fields is discussed. The fractional space generalization of static and time-varying potentials, provided in this chapter, can be used to study electrostatic problems in fractal media. In Sect. 3.1, electrostatic potential in D-dimensional fractional space is studied. In Sect. 3.2, time-varying auxiliary potential is studied in fractional space.

3.1 Electrostatic Potential in Fractional Space In this section, an exact solution of the Laplace’s equation for electrostatic potential in D-dimensional fractional space is presented. As an application, the electrostatic potential inside the rectangular box with surfaces held at constant potentials is obtained in fractional pace. It is also shown that for integer value of dimension D, the classical results are recovered. The obtained solution can be used to study complex electrostatic problems in fractal media. In Sect. 3.1.1, we investigate full analytical solution to the Laplace’s equation in D-dimensional fractional space, where the parameter D is used to describe the measure distribution of space. In Sect. 3.1.2, the electrostatic potential inside the rectangular box with surfaces held at constant potentials in fractional space is obtained. Finally, in Sect. 3.1.3, results are summarized.

3.1.1 An Exact Solution of the Laplace’s Equation in D-dimensional Fractional Space The Laplace’s equation in D-dimensional fractional space can describe complex physical phenomenon. Laplace’s equation in fractional space is given by: 2 ∇D Ψ =0

(3.1.1)

M. Zubair et al., Electromagnetic Fields and Waves in Fractional Dimensional Space, SpringerBriefs in Applied Sciences and Technology, DOI: 10.1007/978-3-642-25358-4_3, © The Author(s) 2012

17

18

3 Potentials for Static and Time-Varying Fields

2 is the scalar laplacian operator in where ψ is the electrostatic potential and ∇D D-dimensional fractional space given as follows [1]. 2 = ∇D

∂2 α1 − 1 ∂ α2 − 1 ∂ α3 − 1 ∂ ∂2 ∂2 + + + + + ∂x2 x ∂x ∂y2 y ∂y ∂z 2 z ∂z

(3.1.2)

where, three parameters (0 < α1 ≤ 1, 0 < α2 ≤ 1 and 0 < α3 ≤ 1) are used to describe the measure distribution of space where each one is acting independently on a single coordinate and the total dimension of the system is D = α1 + α2 + α3 . It is obvious that for three dimensional space (D = 3), if we set α1 = α2 = α3 = 1 2 reduces to the classical Laplacian in (3.1.2), the fractional Laplacian operator ∇D operator ∇ 2 in Euclidean space. In this section we present an exact solution of the Laplace equation in (3.1.2) for electrostatic potential in D-dimensional fractional space. In expanded form, (3.1.1) can be written as:

∂2 α1 − 1 ∂ α2 − 1 ∂ α3 − 1 ∂ ∂2 ∂2 + + + + + 2 2 2 ∂x x ∂x ∂y y ∂y ∂z z ∂z

ψ =0

(3.1.3)

Equation (3.1.6) is separable using separation of variables. We consider ψ(x, y, z) = f (x)g(y)h(z)

(3.1.4)

the resulting ordinary differential equations are obtained as follows: α1 − 1 d d2 2 + + α f (x) = 0 dx2 x dx 2 d α2 − 1 d 2 + β g(y) = 0 + dy2 y dy 2 α3 − 1 d d 2 + − γ h(z) = 0 d z2 z dz

(3.1.5b)

α2 + β 2 = γ 2

(3.1.6)

(3.1.5a)

(3.1.5c)

where, in addition,

Equation (3.1.5a) through (3.1.5c) are all of the same form; solution for any one of them can be replicated for others by inspection.We choose to work first with f (x). We write (3.1.5a) as d d2 + α2 x f = 0 x 2 +a dx dx

(3.1.7)

where, a = α1 − 1. Equation (3.1.7) is reducible to Bessel’s equation under substitution f = xv1 ξ as follows:

3.1 Electrostatic Potential in Fractional Space

d2 d |1 − a| 2 2 2 x +x + (α x − v1 ) ξ = 0, v1 = dx2 dx 2 2

19

(3.1.8)

The solution of Bessel’s equation in (3.1.8) is given as [2] ξ = C1 Jv1 (αx) + C2 Yv1 (αx)

(3.1.9)

where, Jv1 (αx) is referred to as Bessel function of the first kind of order v1 , Yv1 (αx) as the Bessel function of the second kind of order v1 . Finally the solution of (3.1.5a) becomes α1 f (x) = xv1 C1 Jv1 (αx) + C2 Yv1 (αx) , v1 = 1 − (3.1.10) 2 Similarly, the solutions to (3.1.5b) and (3.1.5c) are obtained as α2 (3.1.11) g(y) = yv2 C3 Jv2 (βy) + C4 Yv2 (βy) , v2 = 1 − 2 h(z) = z v3 C5 Jv3 (−jγ z) + C6 Yv3 (−jγ z) α3 = z v3 C5 Jv3 (−j α 2 + β 2 z) + C6 Yv3 (−j α 2 + β 2 z) , v3 = 1 − 2 (3.1.12) From (3.1.4) and (3.1.10) through (3.1.13), the solution of (3.1.3) have the form ψ(x, y, z) = xv1 yv2 z v3 C1 Jv1 (αx) + C2 Yv1 (αx) × C3 Jv2 (βy) + C4 Yv2 (βy) × C5 Jv3 (−j α 2 + β 2 z) + C6 Yv3 (−j α 2 + β 2 z) (3.1.13) where, C1 through C6 are constant coefficients. Also α and β can be determined by imposing boundary conditions on potential. This solution can be used to study the electrostatic field in a non-integer dimensional fractal media.

3.1.2 Electrostatic Potential Inside a Rectangular Box in Fractional Space In the present section we determine the potential inside a rectangular box in D-dimensional fractional space with dimensions (a, b, c) in the (x, y, z) directions. All surfaces of the box are kept at zero potential, except the surface z = c, which is at potential V (x, y). Starting from the requirement that ψ = 0 at x = 0, y = 0, z = 0, it is easy to see that the required forms of f (x), g(y), h(z) are f (x) = xv1 Jv1 (αx)

(3.1.14a)

g(y) = yv2 Jv2 (βy)

(3.1.14b)

20

3 Potentials for Static and Time-Varying Fields

h(z) = z v3 Jv3 (−j α 2 + β 2 z)

(3.1.14c)

To have ψ = 0 at x = a and y = b, we must have αa =

π(4n + 2v1 + 3) 4

(3.1.15a)

βb =

π(4m + 2v2 + 3) 4

(3.1.15b)

With the definitions, αn =

π(4n + 2v1 + 3) 4a

π(4m + 2v2 + 3) 4b 4m + 2v2 + 3 2 π 4n + 2v1 + 3 2 = + 4 a b

(3.1.16a)

βm =

(3.1.16b)

γmn

(3.1.16c)

The partial potential ψmn satisfying all boundary conditions except one, can be written as: ψmn = xv1 yv2 z v3 Jv1 (αn x)Jv2 (βm y)Jv3 (−jγmn z)

(3.1.17)

And the potential can expanded in terms of these ψmn with initial arbitrary coefficients (to be chosen to satisfy final boundary condition): ψ(x, y, z) =

∞

Amn xv1 yv2 z v3 Jv1 (αn x)Jv2 (βm y)Jv3 (−jγmn z)

(3.1.18)

n,m=1

And the final boundary condition ψ = V (x, y) at z = c: V (x, y) =

∞

Amn xv1 yv2 cv3 Jv1 (αn x)Jv2 (βm y)Jv3 (−jγmn c)

(3.1.19)

n,m=1

so, the constant coefficients Amn are given by: a b 4 Amn = dx dyV (x, y)xv1 yv2 Jv1 (αn x)Jv2 (βm y) (3.1.20) abcv3 Jv3 (−jγmn c) 0 0 Equation (3.1.18) provides the required solution in fractional space. Now, if we take D = 3 i.e., α1 = α2 = α3 = 1 in (3.1.18) and use Bessel functions of fractional order then the classical results given by Jackson [3], for the same problem in Euclidean space, can be recovered.

3.1 Electrostatic Potential in Fractional Space

21

3.1.3 Summary An exact solution of the Laplace’s equation for electrostatic potential in D-dimensional fractional space is obtained. The electrostatic potential inside the rectangular box with surfaces held at constant potentials is obtained in fractional space. It is also shown that for integer value of dimension D, the classical results can be recovered. The obtained solution can be used to study complex electrostatic problems in fractal media.

3.2 Time-Varying Potentials in Fractional Space The procedure for analysis of radiation problems is to specify sources and get the fields radiated by the sources. For analysis of radiation problems in fractional space we have to introduce auxiliary potential function A (magnetic vector potential) and F (electric vector potential). The fractional space generalization of the relation between auxiliary potential functions and sources is given by inhomogeneous vector potential wave equations. In this Section, a novel exact solution of the inhomogeneous vector potential wave equations in D-dimensional fractional space is presented. It is also shown that for integer values of dimension D, the classical results are recovered. The solution of inhomogeneous vector potential wave equation in fractional space is useful to study the radiation phenomenon in fractal media. In Sect 3.2.1, we investigate full analytical solution of the inhomogeneous vector potential wave equation in D-dimensional fractional space, where the parameter D is used to describe the measure distribution of space, also the solution of inhomogeneous vector potential wave equation in integer-dimensional space is justified from the results obtained. Finally, results are summarized drawn in Sect. 3.2.2.

3.2.1 Inhomogeneous Vector Potential Wave Equation in D-dimensional Fractional Space The procedure for analysis of radiation problems is to specify sources and get the fields radiated by the sources. For analysis of radiation problems in fractional space we have to introduce auxiliary potential function A (magnetic vector potential) and F (electric vector potential). The fractional space generalization of the relation between auxiliary potential functions and sources is given by inhomogeneous vector potential wave equations as below: 2 A + k 2 A = −μJ ∇D

(3.2.1)

2 ∇D F + k 2 F = −εM

(3.2.2)

22

3 Potentials for Static and Time-Varying Fields

2 is the modified laplacian operator in D-dimensional where, k 2 = ω2 με and ∇D fractional space defined in Equation (3.1.2) and J, M are harmonic electric and magnetic currents. In spherical coordinates fractional Laplacian operator becomes: 2 ∇D

2 ∂ ∂2 D−1 ∂ D−2 ∂ 1 = 2+ + + ∂r r ∂r r2 ∂θ 2 tan θ ∂θ 2 1 D−3 ∂ ∂ + 2 + 2 r sin θ ∂φ tan φ ∂φ

(3.2.3)

where, 2 < D ≤ 3. In this section we present a novel exact solution of the inhomogeneous vector potential wave equations in (3.2.1) and (3.2.2) in D-dimensional fractional space. Once the solution to any one of equation (3.2.1) and (3.2.2) in fractional space is known, the solution to the other can be written by duality principle. To drive the solution to (3.2.1), we assume a source with current density Jz , which in limit is an infinitesimal point source, is placed at origin of a x, y, z coordinate system. Since the current density is directed along z-axis, only an Az component will exist. Thus, using (3.2.1) 2 Az + k 2 Az = −μJz ∇D

(3.2.4)

At point removed from the source (Jz = 0), the wave equation reduces to 2 Az + k 2 Az = 0 ∇D

(3.2.5)

Since in the limit the source is a point, it requires Az as function of r in spherical coordinates (i.e., Az is not a function of θ and φ. Thus, using the definition of Laplacian operator from (3.2.3) we get (3.2.5) in expanded form as:

∂2 D−1 ∂ 2 + + k Az (r) = 0 ∂r2 r ∂r

(3.2.6)

where, 2 < D ≤ 3. The partial derivatives are replaced with ordinary derivative because Az is a function of r only. Equation (3.2.6) is reducible to Bessel’s equation under substitution Az (r) = xn ξ as follows: r2

d2 d D 2 2 2 + r r − n ) ξ = 0, n = 1 − + (k 2 dr dr 2

(3.2.7)

The solution of Bessel’s equation in (3.2.7) is given as [2] ξ = C1 Hn(1) (kr) + C2 Hn(2) (kr) (1)

(3.2.8) (2)

where, Hn (kr) is referred to as Hankel function of the first kind of order n, Hn (kr) as the Hankel function of the second kind of order n. Finally, the solution of (3.2.6) becomes

3.2 Time-Varying Potentials in Fractional Space

23

D Az = rn C1 Hn(1) (kr) + C2 Hn(2) (kr) , n = 1 − 2

(3.2.9)

So the two independent solutions of (3.2.6) are Az1 = C1 rn Hn(1) (kr)

(3.2.10a)

Az2 = C2 rn Hn(2) (kr)

(3.2.10b)

Equations (3.2.10a, b) represent inward and outward going waves (assuming a time dependency ejωt ). For this problem source is located at origin and the waves are going outward. So, we choose the solution of (3.2.6) as Az = C1 rn Hn(2) (kr)

(3.2.11)

In the static case (ω = 0, k = 0), (3.2.11) simplifies to Az = C1 rn Hn(2) (0)

(3.2.12)

Thus, at points removed from the source, the time-varying and static solutions of (3.2.11) and (3.2.12) differ only by the argument of Hankel function from kr to zero; or the time-varying solution can be found by multiplying static form with √ (2) rHn (kr). In the presence of the source (Jz = 0) an (k = 0) the wave equation (2.2.4) is reduced to 2 ∇D Az = −μJz

(3.2.13)

This equation is known as Poisson’s equation in fractional space. The well known form of Poisson’s equation relating the scalar field ψ to the electric charge density ρ is given by 2 ψ= ∇D

ρ ε

and the solution of (3.2.14) is given by [4] as follow: ρ 23−D Γ (3/2) ψ= dV Γ (D/2) rD−2

(3.2.14)

(3.2.15)

where, r is the distance from any point on charge density to observation point. As (3.2.13) is similar to (3.2.14), so its solution is same as in (3.2.15). μ 23−D Γ (3/2) Jz dV (3.2.16) Az = D−2 4π Γ (α/2) r using analogy between (3.2.11) and (3.2.12), the time-varying solution of (3.2.4) is given by

24

3 Potentials for Static and Time-Varying Fields

Az =

μ 23−D Γ (3/2) 4π Γ (D/2)

(2)

Jz

Hn (kr) dV rD−5/2

(3.2.17)

If the current were in the x- and y-directions (Jx ,Jy ), the wave equations for each would reduce to have same form as (3.2.4) and will possess the same solutions as in equation (3.2.17). Finally, we have the solution of vector wave equation in (3.2.4) as: Hn(2) (kr) μ 23−D Γ (3/2) J D−5/2 d V A= (3.2.18) 4π Γ (D/2) r If the source is removed from origin and placed at position represented by the primed coordinates (x , y , z ), (3.2.18) can be written as: (2) μ 23−D Γ (3/2) Hn (kR) A= (3.2.19) J(x , y , z ) D−5/2 d V 4π Γ (D/2) R where, R is the distance between any point in the source to observation point, Γ (x) (2) is the gamma function and Hn (kR) denotes the Hankel function of second kind of order n representing outward waves from source point. Now, for validation of our provided solution in (3.2.19), we get vector potential A from our solution by substituting D = 3 in (3.2.19). For D = 3, we have n = −1/2. Using following Hankel function of fractional order [5] 2 −jx 2 (3.2.20) e H 1 (x) = πx 2 Equation (3.2.19) gets reduced to e−jkR μ A = C dV J(x , y , z ) 4π R

(3.2.21)

where, C is a constant term. Equation (3.2.21) is in exact agreement with the solution provided in [6] for Euclidean space. In a similar fashion, we can show the solution of (3.2.2) as (2) ε 23−D Γ (3/2) Hn (kR) F= (3.2.22) M(x , y , z ) D−5/2 d V 4π Γ (D/2) R For D = 3, using Hankel function of fractional order given by (3.2.20), finally equation (3.2.22) gets reduced to e−jkR ε (3.2.23) dV M(x , y , z ) F=C 4π R Equation (3.2.23) is in exact agreement with the solution provided in [6] for Euclidean space. The fractional space solutions of inhomogeneous vector potential wave equation, given by (3.2.19) and (3.2.22), can be used to solve complex radiation problems in fractional space.

3.2 Time-Varying Potentials in Fractional Space

25

3.2.2 Summary An exact solution of the inhomogeneous vector potential wave equation in D-dimensional fractional space is presented. It is also shown that for integer values of dimension D, the classical results are recovered. The solution of inhomogeneous vector potential wave equation in fractional space is useful to study the radiation phenomenon in fractal media.

References 1. C. Palmer, P.N. Stavrinou, Equations of motion in a noninteger-dimension space. J. Phys. A 37, 6987–7003 (2004) 2. A.D. Polyanin, V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. (CRC Press, Boca Raton, 2003) 3. J.D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley, New York, 1999) 4. S. Muslih, D. Baleanu, Fractional Multipoles in fractional space. Nonlinear Anal. Real World Appl. 8, 198–203 (2007) 5. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables (U.S. Department of Commerce, USA, 1972) 6. C.A. Balanis, Antenna Theory: Analysis and Design (Wiley, New York, 1982)

Chapter 4

Electromagnetic Wave Propagation in Fractional Space

The wave equation has very important role in many areas of physics. It has a fundamental meaning in classical as well as quantum field theory. With this view, one is strongly motivated to discuss solutions of the wave equation in all possible situations. The wave equation in fractional space can effectively describe the wave propagation phenomenon in fractal media. In this chapter, exact solutions of different forms of wave equation in D-dimensional fractional space are provided, which describe the phenomenon of electromagnetic wave propagation in fractional space. In Sect. 4.1, the general plane wave solutions in fractional space are provided for lossless medium case. In Sect. 4.2, the general plane wave solutions in fractional space are provided for lossy medium case. In Sect. 4.3, the general cylindrical wave solutions in fractional space are provided by solving cylindrical wave equation in D-dimensional fractional space. In Sect. 4.4, the general spherical wave solutions in fractional space are provided by solving spherical wave equation in D-dimensional fractional space.

4.1 General Plane Wave Solutions in Fractional Space: Lossless Medium Case In Sect. 4.1.1, we investigate full analytical solution of wave equation in D-dimensional fractional space, where three parameters are used to describe the measure distribution of space. In Sect. 4.1.2, solution of wave equation in integerdimensional space is obtained from the results of previous section. Finally, in Sect. 4.1.3, major results are summarized.

4.1.1 General Plane Wave Solutions in Fractional Space For source-free and lossless media, the vector wave equations for the complex electric and magnetic field intensities are given by the Helmholtz’s equation as follows [1]. M. Zubair et al., Electromagnetic Fields and Waves in Fractional Dimensional Space, SpringerBriefs in Applied Sciences and Technology, DOI: 10.1007/978-3-642-25358-4_4, © The Author(s) 2012

27

28

4 Electromagnetic Wave Propagation in Fractional Space 2 ∇D E + β2E = 0

(4.1.1a)

+β H =0

(4.1.1b)

2 H ∇D

2

where, β 2 = ω2 με. Time dependency e jwt has been suppressed throughout the 2 is the scalar Laplacian operator in D dimensional fractional discussion. Here ∇ D space and is defined as follows [2]. 2 ∇D =

∂2 ∂2 ∂2 α1 − 1 ∂ α2 − 1 ∂ α3 − 1 ∂ + + + + + 2 2 2 ∂x x ∂x ∂y y ∂y ∂z z ∂z

(4.1.2)

Eq. (4.1.2) uses three parameters (0 < α1 ≤ 1, 0 < α2 ≤ 1 and 0 < α3 ≤ 1) to describe the measure distribution of space where each one is acting independently on a single coordinate and the total dimension of the system is D = α1 + α2 + α3 . Once the solution to any one of Eqs. (4.1.1a and 4.1.1b) in fractional space is known, the solution to the other can be written by an interchange of E with H or H with E due to duality. We will examine the solution for E. In rectangular coordinates, a general solution for E can be written as E(x, y, z) = aˆ x E x (x, y, z) + aˆ y E y (x, y, z) + aˆ z E z (x, y, z)

(4.1.3)

Substituting (4.1.3) into (4.1.1a) we can write that 2 ∇D (aˆ x E x + aˆ y E y + aˆ z E z ) + β 2 (aˆ x E x + aˆ y E y + aˆ z E z ) = 0

(4.1.4)

which reduces to three scalar wave equations as follows: 2 E x (x, y, z) + β 2 E x (x, y, z) = 0 ∇D

(4.1.5a)

2 E y (x, ∇D 2 ∇D E z (x,

y, z) + β E y (x, y, z) = 0

(4.1.5b)

y, z) + β 2 E z (x, y, z) = 0

(4.1.5c)

2

Eq. (4.1.5a) through (4.1.5c) are all of the same form; solution for any one of them in fractional space can be replicated for others by inspection.We choose to work first with E x as given by (4.1.5a). In expanded form (4.1.5a) can be written as α1 − 1 ∂ E x α2 − 1 ∂ E x ∂ 2 Ex ∂ 2 Ex ∂2 Ex + + + + 2 2 ∂x x ∂x ∂y y ∂y ∂z 2 α3 − 1 ∂ E x + + β2 Ex = 0 z ∂z

(4.1.6)

Equation (4.1.6) is separable using separation of variables. We consider E x (x, y, z) = f (x)g(y)h(z) the resulting ordinary differential equations are obtained as follows:

(4.1.7)

4.1 General Plane Wave Solutions in Fractional Space: Lossless Medium Case

29

d2 α1 − 1 d 2 + + βx f = 0 dx2 x dx 2 d α2 − 1 d 2 + + β y g =0 dy 2 y dy 2 d α3 − 1 d 2 + β + z h =0 dz 2 z dz

(4.1.8b)

βx2 + β y2 + βz2 = β 2

(4.1.9)

(4.1.8a)

(4.1.8c)

where, in addition,

Equation (4.1.9) is referred to as constraint equation. In addition βx , β y , βz are known as wave constants in the x, y, z directions, respectively, which will be determined using boundary conditions. Equation (4.1.8a) through (4.1.8c) are all of the same form; solution for any one of them can be replicated for others by inspection.We choose to work first with f (x). We write (4.1.8a) as x

d d2 +a + βx2 x 2 dx dx

f =0

(4.1.10)

where, a = α1 − 1. Equation (4.1.10) is reducible to Bessel’s equation under substitution f = x n ξ as follows:

d |1 − a| d2 2 2 2 +x + (βx x − n ) ξ = 0, n = x 2 dx dx 2 2

(4.1.11)

The solution of Bessel’s equation in (4.1.11) is given as [3] ξ = C1 Jn (βx x) + C2 Yn (βx x)

(4.1.12)

where, Jn (βx x) is referred to as Bessel function of the first kind of order n, Yn (βx x) as the Bessel function of the second kind of order n. Finally the solution of (4.1.8a) becomes α1 f (x) = x n 1 C1 Jn 1 (βx x) + C2 Yn 1 (βx x) , n 1 = 1 − 2

(4.1.13)

Similarly, the solutions to (4.1.8b) and (4.1.8c) are obtained as α2 g(y) = y n 2 C3 Jn 2 (β y y) + C4 Yn 2 (β y y) , n 2 = 1 − 2 α3 n3 h(z) = z C5 Jn 3 (βz z) + C6 Yn 3 (βz z) , n 3 = 1 − 2

(4.1.14) (4.1.15)

From (4.1.7) and (4.1.13) through (4.1.15), the solution of (4.1.5a) have the form

30

4 Electromagnetic Wave Propagation in Fractional Space

E x (x, y, z) = x n 1 y n 2 z n 3 C1 Jn 1 (βx x) + C2 Yn 1 (βx x) × C 3 Jn 2 (β y y) + C 4 Yn 2 (β y y) × C5 Jn 3 (βz z) + C 6 Yn 3 (βz z)

(4.1.16)

where, C1 through C 6 are constant coefficients. Similarly, the solutions to (4.1.5b) and (4.1.5c) are obtained as E y (x, y, z) = x n 1 y n 2 z n 3 D1 Jn 1 (βx x) + D2 Yn 1 (βx x) × D3 Jn 2 (β y y) + D4 Yn 2 (β y y) (4.1.17) × D5 Jn 3 (βz z) + D6 Yn 3 (βz z) and

E z (x, y, z) = x n 1 y n 2 z n 3 F1 Jn 1 (βx x) + F2 Yn 1 (βx x) × F3 Jn 2 (β y y) + F4 Yn 2 (β y y) × F5 Jn 3 (βz z) + F6 Yn 3 (βz z)

(4.1.18)

where, D1 through D6 and F1 through F6 are constant coefficients. For e jwt time variations, the instantaneous form E (x, y, z; t) of the vector complex function E(x, y, z) in (4.1.8a) takes the form E (x, y, z; t) = e[{aˆ x E x (x, y, z) + aˆ y E y (x, y, z) + aˆ z E z (x, y, z)}e jwt ] (4.1.19) where E x (x, y, z), E y (x, y, z) and E z (x, y, z) are given by (4.1.16) through (4.1.18). Equation (4.1.19) provides a general plane wave solution in fractional space. This solution can be used to study the phenomenon of electromagnetic wave propagation in any non-integer dimensional space.

4.1.2 Discussion on Fractional Space Solution Equation (4.1.19) is the generalization of the concept of wave propagation in integer dimensional space to the wave propagation in non-integer dimensional space. As a special case, for three-dimensional space, this problem reduces to classical wave propagation concept; i.e., if we set α1 = 1 in Eq. (4.1.13) then n 1 = 12 and it gives 1 (4.1.20) f (x) = x 2 C1 J 1 (βx x) + C2 Y 1 (βx x) 2

2

Using Bessel functions of fractional order [4]: 2 J 1 (x) = sin (x) 2 πx 2 cos (x) Y 1 (x) = − 2 πx

(4.1.21a) (4.1.21b)

4.1 General Plane Wave Solutions in Fractional Space: Lossless Medium Case

31

Eq. (4.1.13) can be reduced to f (x) = C1 sin(βx x) + C2 cos(βx x)

(4.1.22)

where, Ci = Ci πβ2 x , i = 1, 2 Similarly, we set α2 = 1 and α3 = 1 in (4.1.14) and (4.1.15) respectively and using Bessel functions of fractional order in (4.1.21a) through (b), we get g(y) = C3 sin(β y y) + C 4 cos(β y y) h(z) = C 5 sin(βz z) + C6 cos(βz z)

(4.1.23) (4.1.24)

From (4.1.22) through (4.1.24), we get E x (x, y, z) in three-dimensional space (D = 3) as follows E x (x, y, z) = C1 sin(βx x) + C2 cos(βx x) × C 3 sin(β y y) + C 4 cos(β y y) × C 5 sin(βz z) + C6 cos(βz z) (4.1.25) which is comparable to the solution of wave equation in integer dimensional space obtained by Balanis [1]. Similarly, field components E y (x, y, z) and E z (x, y, z) can also be reduced for three-dimensional case. As another special case, if we choose a single parameter for non-integer dimension D where 2 < D ≤ 3, i.e., we take α1 = α2 = 1 so D = α3 + 2. In this case from Eq. (4.1.6) we obtain ∂ 2 Ex ∂2 Ex D − 3 ∂ Ex ∂ 2 Ex + + + + β2 Ex = 0 ∂x2 ∂ y2 ∂z 2 z ∂z

(4.1.26)

Solving this equation by separation of variables leads to the following result E x (x, y, z) = z n [G 1 cos(βx x) + G 2 sin(βx x)] × G 3 cos(β y y) + G 4 sin(β y y) × G 5 Jn (βz z) + G 6 Yn (βz z) where, n = 2 −

D 2.

(4.1.27)

Here if we set D = 3, and using (4.1.21a) and (b), we get

E x (x, y, z) =

2 [G 1 cos(βx x) + G 2 sin(βx x)] πβz × G 3 cos(β y y) + G 4 sin(β y y) × G 5 sin(βz z) + G 6 cos(βz z)

(4.1.28)

where, G 1 through G 6 are constant coefficients. The result obtained in (4.1.28) is comparable to that obtained by Balanis [1] for 3-dimensional space.

32

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.1 Usual wave propagation (D = 3). [This figure was originally published in [5] , reproduced courtesy of The Electromagnetics Academy]

As an example, an infinite sheet of surface current can be considered as a source of plane waves in D-dimensional fractional space. We assume that an infinite sheet of electric surface current density Js = Js0 xˆ exists on the z = 0 plane in free space. Since the sources do not vary with x or y, the fields will not vary with x or y but will propagate away from the source in ±z direction. The boundary conditions to be satisfied at z = 0 are zˆ × (E2 − E1 ) = 0 and zˆ × (H2 − H1 ) = Js0 x, ˆ where E1 , H1 are the fields for z < 0, and E2 , H2 are the fields for z > 0. To satisfy the later boundary condition, H must have a yˆ component. Then for E to be normal to H and zˆ , E must have an xˆ component. Thus, the corresponding wave equation for E and H fields in D-dimensional fractional space where 2 < D ≤ 3 can be written by modifying (4.1.26) as d2 Ex D − 3 d Ex + + β2 Ex = 0 dz 2 z dz d 2 Hy D − 3 d Hy + + β 2 Hy = 0 2 dz z dz

(4.1.29a) (4.1.29b)

Solution of (4.1.29a) and (4.1.29b) takes the similar form as (4.1.27) and under above mentioned boundary conditions the fields will have the following form: Js0 Js Jn (βz z), H1 = yˆ z n 0 Jn (βz z); z < 0 2 2η0 n Js0 n Js0 Yn (βz z); z > 0 Yn (βz z), H2 = − yˆ z E2 = −xˆ z 2 2η0 E1 = −xˆ z n

(4.1.30a) (4.1.30b)

4.1 General Plane Wave Solutions in Fractional Space: Lossless Medium Case

33

Fig. 4.2 Wave propagation in fractional space (D = 2.5). [This figure was originally published in [5] , reproduced courtesy of The Electromagnetics Academy]

where, η0 is wave impedance in free space. Assuming a time dependency e jwt and Js0 = −2 A/m, the solution for the usual wave for z > 0 with D = 3 is shown in Fig. 4.1, which is comparable to well known plane wave solutions in 3-dimensional space [1]. Similarly, for D = 2.5 we have fractal medium wave for z > 0 as shown in Fig. 4.2, where amplitude variations are described in terms of Bessel functions.

4.1.3 Summary General plane wave solution in source-free and lossless media in fractional space is presented by solving vector wave equation in D-dimensional fractional space. When the wave propagates in fractional space, the amplitude variations are described by Bessel functions. The obtained general plane wave solution is a generalization of integer-dimensional solution to a non-integer dimensional space. For all investigated cases when D is an integer-dimension, the classical results are recovered.

4.2 General Plane Wave Solutions in Fractional Space: Lossy Medium Case In this section, an extension of previous work to the case of plane wave propagation in lossy medium in fractional space is presented. The generalized analytical solution investigated in this section have potential applications in electromagnetic wave propagation problems in lossy media present in fractional space.

34

4 Electromagnetic Wave Propagation in Fractional Space

In Sect. 4.2.1 we investigate full analytical solution of Helmholtz’s equation in source-free, lossy media present in D-dimensional fractional space, where three parameters are used to describe the measure distribution of space. In Sect. 4.2.2, solution of wave equation in integer-dimensional space is obtained from the results of previous section. In Sect. 4.2.3, an example of wave propagation in lossy medium due to current sheet as source of plane waves in fractional space is presented. Finally, results are summarized in Sect. 4.2.4.

4.2.1 General Plane Wave Solutions in Lossy Medium in Fractional Space For source-free and lossy media, the vector wave equations for the complex electric and magnetic field intensities are given by the Helmholtz’s equation as follows [1]. 2 E − γ 2E = 0 ∇D 2 ∇D H − γ 2H = 0

(4.2.1) (4.2.2)

γ 2 = jωμ(σ + jωε)

(4.2.3)

γ = α + jβ σ 2 με

α=ω 1+ −1 2 ωε σ 2 με

1+ +1 β=ω 2 ωε

(4.2.4)

where

(4.2.5)

(4.2.6)

In equation (4.2.3) and (4.2.4) γ = propagation constant α = attenuation constant (Np/m) β = phase constant (rad/m) ε = permittivity of medium (H/m) μ = permeability of medium (F/m) σ = conductivity of medium (S/m) Time dependency e jwt has been suppressed throughout the discussion. 2 is the Laplacian operator in D-dimensional fractional In Eqs. (4.2.1) and (4.2.2), ∇ D space and is defined as follows [2]. 2 = ∇D

∂2 ∂2 ∂2 α1 − 1 ∂ α2 − 1 ∂ α3 − 1 ∂ + + + + + ∂x2 x ∂x ∂ y2 y ∂y ∂z 2 z ∂z

(4.2.7)

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

35

Equation (4.2.7) uses three parameters (0 < α1 ≤ 1, 0 < α2 ≤ 1 and 0 < α3 ≤ 1) to describe the measure distribution of space where each one is acting independently on a single coordinate and the total dimension of the system is D = α1 + α2 + α3 . Once the solution to any one of Eq. (4.2.1) and (4.2.2) in fractional space is known, the solution to the other can be written by an interchange of E with H or H with E due to duality. We will examine the solution for E. In rectangular coordinates, a general solution for E can be written as E(x, y, z) = aˆ x E x (x, y, z) + aˆ y E y (x, y, z) + aˆ z E z (x, y, z)

(4.2.8)

Substituting (4.2.8) into (4.2.1) we can write that 2 (aˆ x E x + aˆ y E y + aˆ z E z ) − γ 2 (aˆ x E x + aˆ y E y + aˆ z E z ) = 0 ∇D

(4.2.9)

which reduces to following three scalar wave equations 2 ∇D E x (x, y, z) − γ 2 E x (x, y, z) = 0

(4.2.10)

2 E y (x, y, z) − γ 2 E y (x, y, z) = 0 ∇D

(4.2.11)

2 E z (x, y, z) − γ 2 E z (x, y, z) = 0 ∇D

(4.2.12)

Equation (4.2.10) through (4.2.12) are all of the same form; solution for any one of them in fractional space can be replicated for others by inspection.We choose to work first with E x as given by (4.2.10). In expanded form (4.2.10) can be written as ∂ 2 Ex ∂2 Ex α1 − 1 ∂ E x α2 − 1 ∂ E x + + + ∂x2 x ∂x ∂ y2 y ∂y 2 ∂ Ex α3 − 1 ∂ E x + − γ 2 Ex = 0 + 2 ∂z z ∂z

(4.2.13)

Equation (4.2.13) is separable using separation of variables. We consider E x (x, y, z) = f (x)g(y)h(z)

(4.2.14)

the resulting ordinary differential equations are obtained as follows:

α1 − 1 d d2 2 + − γ f =0 x dx2 x dx 2 α1 − 1 d d + − γ y2 g = 0 2 dy y dy 2 d α1 − 1 d 2 − γ + z h =0 dz 2 z dz

where, in addition,

(4.2.15) (4.2.16) (4.2.17)

36

4 Electromagnetic Wave Propagation in Fractional Space

γx2 + γ y2 + γz2 = γ 2

(4.2.18)

Equation (4.2.18) is referred to as constraint equation. In addition γx , γ y , γz are known as wave constants in the x, y, z directions, respectively, that will be determined using boundary conditions. Equation (4.2.15) through (4.2.17) are all of the same form; solution for any one of them can be replicated for others by inspection.We choose to work first with f (x). We write (4.2.15) as

d d2 − γx2 x x 2 +a dx dx

f =0

(4.2.19)

where, a = α1 − 1. Equation (4.2.19) is reducible to Bessel’s equation under substitution f = x n ζ as follows:

d |1 − a| d2 2 2 2 +x + (−γx x − n ) ζ = 0, n = x dx2 dx 2 2

(4.2.20)

The solution of Bessel’s equation in (4.2.20) is given as [3] ζ = C 1 Jn (− jγx x) + C2 Yn (− jγx x)

(4.2.21)

where, Jn (− jγx x) is referred to as Bessel function of the first kind of order n, Yn (− jγx x) as the Bessel function of the second kind of order n. Finally, the solution of (4.2.15) becomes α1 f (x) = x n 1 C1 Jn 1 (− jγx x) + C2 Yn 1 (− jγx x) , n 1 = 1 − 2

(4.2.22)

Similarly, the solutions to (4.2.16) and (4.2.17) are obtained as α2 g(y) = y n 2 C3 Jn 2 (− jγ y y) + C4 Yn 2 (− jγ y y) , n 2 = 1 − 2 α3 n3 h(z) = z C5 Jn 3 (− jγz z) + C6 Yn 3 (− jγz z) , n 3 = 1 − 2

(4.2.23) (4.2.24)

From (4.2.14) and (4.2.22) through (4.2.24), the solution of (4.2.10) for E x (x, y, z) in D-dimensional fractional space have the form E x (x, y, z) = x n 1 y n 2 z n 3 C1 Jn 1 (− jγx x) + C2 Yn 1 (− jγx x) × C3 Jn 2 (− jγ y y) + C4 Yn 2 (− jγ y y) × C 5 Jn 3 (− jγz z) + C 6 Yn 3 (− jγz z) (4.2.25) where, C1 through C6 are constant coefficients.

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

Similarly, the solutions to (4.2.11) and (4.2.12) are obtained as E y (x, y, z) = x n 1 y n 2 z n 3 D1 Jn 1 (− jγx x) + D2 Yn 1 (− jγx x) × D3 Jn 2 (− jγ y y) + D4 Yn 2 (− jγ y y) × D5 Jn 3 (− jγz z) + D6 Yn 3 (− jγz z) and

E z (x, y, z) = x n 1 y n 2 z n 3 F1 Jn 1 (− jγx x) + F2 Yn 1 (− jγx x) × F3 Jn 2 (− jγ y y) + F4 Yn 2 (− jγ y y) × F5 Jn 3 (− jγz z) + F6 Yn 3 (− jγz z)

37

(4.2.26)

(4.2.27)

where, D1 through D6 and F1 through F6 are constant coefficients. For e jwt time variations, the instantaneous form E (x, y, z; t) of the vector complex function E(x, y, z) in (4.2.8) takes the form E (x, y, z; t) = e[{aˆ x E x (x, y, z) + aˆ y E y (x, y, z) +aˆ z E z (x, y, z)}e jwt ]

(4.2.28)

where, E x (x, y, z), E y (x, y, z) and E z (x, y, z) are given by (4.2.25) through (4.2.27). Equation (4.2.28) provides a general plane wave solution for lossy media in fractional space. This solution can be used to study the phenomenon of electromagnetic wave propagation in any non-integer dimensional space.

4.2.2 Discussion on Fractional Space Solution in Lossy Medium Equation (4.2.28) is the generalization of solution for Helmholtz’s equation in integer dimensional space to a non-integer dimensional space. As a special case, for threedimensional space, this problem reduces to classical wave propagation concept; i.e., if we set α1 = 1 in Eq. (4.2.22) then n 1 = 12 and it gives 1 f (x) = x 2 C 1 J 1 (− jγx x) + C2 Y 1 (− jγx x) (4.2.29) 2

2

Using Bessel functions of fractional order [4]: 2 sin (x) J 1 (x) = 2 πx 2 cos (x) Y 1 (x) = − 2 πx Eq. (4.2.22) can be reduced to

(4.2.30) (4.2.31)

38

4 Electromagnetic Wave Propagation in Fractional Space

f (x) =

2j C1 sinh(γx x) + C2 cosh(γx x) π γx

(4.2.32)

Similarly, we set α2 = 1 and α3 = 1 in (4.2.23) and (4.2.24) respectively and using Bessel function of fractional order in (4.2.30) through (4.2.31), we get

2j (4.2.33) C3 sinh(γ y y) + C4 cosh(γ y y) g(y) = π γy

h(z) =

2j C5 sinh(γz z) + C 6 cosh(γz z) π γz

(4.2.34)

From (4.2.32) through (4.2.34), we get E x (x, y, z) in D = 3 dimensional space as follows

8 C1 sinh(γx x) + C2 cosh(γx x) E x (x, y, z) = jπ 3 γx γ y γz × C 3 sinh(γ y y) + C4 cosh(γ y y) (4.2.35) × C5 sinh(γz z) + C 6 cosh(γz z) Similarly, for D = 3 Eqs. (4.2.26) and (4.2.27) can be reduced to

8 D1 sinh(γx x) + D2 cosh(γx x) E y (x, y, z) = 3 jπ γx γ y γz × D3 sinh(γ y y) + D4 cosh(γ y y) × D5 sinh(γz z) + D6 cosh(γz z)

E z (x, y, z) =

8 jπ 3 γ

x γ y γz

F1 sinh(γx x) + F2 cosh(γx x)

(4.2.36)

× F3 sinh(γ y y) + F4 cosh(γ y y) × F5 sinh(γz z) + F6 cosh(γz z)

(4.2.37)

Now, solution obtained in (4.2.35) through (4.2.37) is comparable to the well known solution of Helmholtz’s equation in lossy media present in integer dimensional space obtained by Balanis [1]. As a special case, if we choose a single parameter for non-integer dimension D where 2 < D ≤ 3, i.e., we take α1 = α2 = 1 so D = α3 + 2. In this case from Eq. (4.2.13) we obtain ∂2 Ex ∂2 Ex ∂2 Ex D − 3 ∂ Ex − γ 2 Ex = 0 + + + ∂x2 ∂ y2 ∂z 2 z ∂z

(4.2.38)

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

39

Solving this equation by separation of variables leads to the following result E x (x, y, z) = z n G 1 cosh(γx x) + G 2 sinh(γx x) × G 3 cosh(γ y y) + G 4 sinh(γ y y) (4.2.39) × G 5 Jn (− jγz z) + G 6 Yn (− jγz z) or E x (x, y, z) = z n G 1 cosh(γx x) + G 2 sinh(γx x) × G 3 cosh(γ y y) + G 4 sinh(γ y y) × G 5 Hn(1) (− jγz z) + G 6 Hn(2) (− jγz z)

(4.2.40)

In (4.2.39) and (4.2.40), n = 2 − D2 . Also G 1 through G 6 and G 5 through G 6 are constant coefficients. In (4.2.39) Jn (− jγz z) is referred to as Bessel function of the first kind of order n, Yn (− jγz z) as the Bessel function of the second kind of order n (1) and both are used to represent standing waves. In (4.2.40) Hn (− jγz z) is referred to (2) as Hankel function of the first kind of order n, Hn (− jγz z) as the Hankel function of the second kind of order n and both are used to represent traveling waves. Now, as a special case of fractional space solution, if we set D = 3 in (4.2.39), and use (4.2.30) and (4.2.31), we get

E x (x, y, z) =

2j G 1 cosh(γx x) + G 2 sinh(γx x) π γz × G 3 cosh(γ y y) + G 4 sinh(γ y y) × G 5 sinh(γz z) + G 6 cosh(γz z)

(4.2.41)

The result obtained in (4.2.41) is comparable to well know integer dimensional solution of wave equation obtained by Balanis [1]. Finally, if we take α = 0 in (4.2.4) then using γ = jβ in (4.2.25) through (4.2.27), we can reduce the fractional space solution for lossy medium to fractional space solution for lossless medium as below: E x (x, y, z) = x n 1 y n 2 z n 3 C1 Jn 1 (βx x) + C2 Yn 1 (βx x) × C 3 Jn 2 (β y y) + C4 Yn 2 (β y y) (4.2.42) × C5 Jn 3 (βz z) + C6 Yn 3 (βz z) E y (x, y, z) = x n 1 y n 2 z n 3 D1 Jn 1 (βx x) + D2 Yn 1 (βx x) × D3 Jn 2 (β y y) + D4 Yn 2 (β y y) × D5 Jn 3 (βz z) + D6 Yn 3 (βz z)

(4.2.43)

40

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.3 Current sheet as source of plane waves in fractional space

E z (x, y, z) = x n 1 y n 2 z n 3 F1 Jn 1 (βx x) + F2 Yn 1 (βx x) × F3 Jn 2 (β y y) + F4 Yn 2 (β y y) × F5 Jn 3 (βz z) + F6 Yn 3 (βz z)

(4.2.44)

Fractional space solution for lossless medium obtained in (4.2.42) through (4.2.44) is in exact agreement with that obtained in previous section.

4.2.3 Example: Current Sheet as Source of Plane Waves in Fractional Space As an example, an infinite sheet of surface current can be considered as a source of plane waves in lossy medium present in D-dimensional fractional space. We assume that an infinite sheet of electric surface current density Js = Js0 xˆ exists on the z = 0 plane in lossy medium characterized by ε1 , μ1 , σ1 in half-space z < 0 and by ε2 , μ2 , σ2 in half-space z > 0 (see Fig. 4.3). Since the sources do not vary with x or y, the fields will not vary with x or y but will propagate away from the source in ±z direction. The boundary conditions to be satisfied at z = 0 are zˆ × (E2 − E1 ) = 0 and zˆ × (H2 − H1 ) = Js0 x, ˆ where E1 , H1 are the fields for z < 0, and E2 , H2 are the fields for z > 0. To satisfy the later boundary condition, H must have a yˆ component. Then for E to be normal to H and zˆ , E must have an xˆ component. Thus, the corresponding wave equation for E and H fields in D-dimensional fractional space where 2 < D ≤ 3 can be written by modifying (4.2.38) as

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

41

Fig. 4.4 Usual wave propagation (D = 3) in lossy medium

d2 Ex D − 3 d Ex + − γ 2 Ex = 0 2 dz z dz d 2 Hy D − 3 d Hy + − γ 2 Hy = 0 2 dz z dz

(4.2.45) (4.2.46)

Solution of (4.2.45) and (4.2.46) takes the similar form as (4.2.40) and under above mentioned boundary conditions the fields will have the following form: Js0 n (1) z Hn (− jγ1 z); 2 Js H1 = yˆ 0 z n Hn(1) (− jγ1 z); 2η1 Js E2 = −xˆ 0 z n Hn(2) (− jγ2 z); 2 Js0 n (2) z Hn (− jγ2 z); H2 = − yˆ 2η2 E1 = −xˆ

z0

(4.2.50)

jωμ2 1 where, γ12 = jωμ1 (σ1 + jωε1 ), γ22 = jωμ2 (σ2 + jωε2 ), η1 = jωμ γ1 , η2 = γ2 . Assuming a time dependency e jwt and Js0 = −2 A/m, the solution for the usual wave for z > 0 with D = 3 is shown in Fig. 4.4, which is comparable to well known plane wave solutions for 3-dimensional space in [1]. Similarly, for D = 2.5 we have fractal wave propagating in fractional space for z > 0 as shown in Fig. 4.5, where amplitude variations are described in terms of Hankel functions.

42

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.5 Wave propagation in lossy medium present in fractional space (D = 2.5)

4.2.4 Summary The phenomenon of wave propagation in source-free and lossy media in fractional space is studied by solving Helmholtz’s equation in D-dimensional fractional space. When the wave propagates in lossy media in the fractional space, the amplitude variations are described by Bessel functions. For all investigated cases when D is an integer-dimension, the classical results are recovered. The plane wave solutions investigated in this paper have potential applications in wave propagation problems in fractional space.

4.3 Cylindrical Wave Propagation in Fractional Space In this section, we present an exact solution of cylindrical wave equation in fractional space that can be used to describe the phenomenon of wave propagation in any fractal media. In Sect. 4.3.1, we investigate full analytical cylindrical wave solution to the wave equation in D-dimensional fractional space, where the parameter D is used to describe the measure distribution of space. In Sect. 4.3.2, the solution of wave equation in integer-dimensional space is justified from the results of previous section. Finally, in Sect. 4.3.3, major results are summarized.

4.3 Cylindrical Wave Propagation in Fractional Space

43

4.3.1 An Exact Solution of Cylindrical Wave Equation in Fractional Space The problems that exhibit cylindrical geometries are needed to be solved using cylindrical coordinate system. As for the case of rectangular geometries, the electric and magnetic fields of cylindrical geometry boundary-value problem must satisfy corresponding cylindrical wave equation [1]. Let us assume that the space in which fields must be solved is fractional dimensional and source-free. For source-free and lossless media, the vector wave equations for the complex electric and magnetic field intensities are given by the Helmholtz’s equation as follows [1]. 2 ∇D E + β2E = 0

(4.3.1)

2 H + β2H = 0 ∇D

(4.3.2)

where, β 2 = ω2 με. Time dependency e jwt has been suppressed throughout the 2 is the Laplacian operator in D-dimensional fractional space discussion. Here, ∇ D and is defined in rectangular coordinate system as follows [2]. 2 ∇D =

∂2 α1 − 1 ∂ α2 − 1 ∂ α3 − 1 ∂ ∂2 ∂2 + + + + + 2 2 2 ∂x x ∂x ∂y y ∂y ∂z z ∂z

(4.3.3)

where x, y and z are rectangular coordinates. Equation (4.3.3) uses three parameters (0 < α1 ≤ 1, 0 < α2 ≤ 1 and 0 < α3 ≤ 1) to describe the measure distribution of space where each one is acting independently on a single coordinate and the total dimension of the system is D = α1 + α2 + α3 . To find cylindrical wave solutions of wave equation in D-dimensional fractional space, it is likely that a cylindrical coordinate system (ρ, φ, z) will be used. In cylindrical coordinate system (4.3.3) becomes 2 ∇D =

∂2 1 ∂ + (α1 + α2 − 1) ∂ρ 2 ρ ∂ρ 2 1 ∂ ∂ + 2 − {(α1 − 1) tan φ − (α2 − 1) cot φ} ρ ∂φ 2 ∂φ +

∂2 α3 − 1 ∂ + 2 ∂z z ∂z

(4.3.4)

Once the solution to any one of Eqs. (4.3.1) and (4.3.2) in fractional space is known, the solution to the other can be written by an interchange of E with H or H with E due to duality [1]. We will examine the solution for E. In cylindrical coordinates, a general solution for E can be written as E(ρ, φ, z) = aˆ ρ E ρ (ρ, φ, z) + aˆ φ E φ (ρ, φ, z) + aˆ z E z (ρ, φ, z) Substituting (4.3.5) into (4.3.1) we can write that

(4.3.5)

44

4 Electromagnetic Wave Propagation in Fractional Space 2 ∇D (aˆ ρ E ρ + aˆ φ E φ + aˆ z E z ) + β 2 (aˆ ρ E ρ + aˆ φ E φ + aˆ z E z ) = 0

(4.3.6)

Since, 2 2 (aˆ ρ E ρ ) = aˆ ρ ∇ D Eρ ∇D

(4.3.7)

2 2 ∇D (aˆ φ E φ ) = aˆ φ ∇ D Eφ

(4.3.8)

2 2 (aˆ z E z ) = aˆ z ∇ D Ez ∇D

(4.3.9)

So, Eq. (4.3.6) cannot be reduced to simple scalar wave equations, but it can be reduced to coupled scalar partial differential equations. However for simplicity, the wave mode solution can be formed in cylindrical coordinates that must satisfy the following scalar wave equation: 2 ψ(ρ, φ, z) + β 2 ψ(ρ, φ, z) = 0 ∇D

(4.3.10)

where, ψ(ρ, φ, z) is a scalar function that can represent a field or vector potential component. In expanded form (4.3.10) can be written as ∂ 2ψ 1 ∂ψ + (α1 + α2 − 1) ∂ρ 2 ρ ∂ρ 2 1 ∂ψ ∂ψ + 2 − {(α − 1) tan φ − (α − 1) cot φ} 1 2 ρ ∂φ 2 ∂φ +

∂ 2ψ α3 − 1 ∂ψ + + β 2ψ = 0 2 ∂z z ∂z

(4.3.11)

Equation (4.3.11) is separable using method of separation of variables. We consider ψ(ρ, φ, z) = f (ρ)g(φ)h(z)

(4.3.12)

the resulting ordinary differential equations are obtained as follows: 2 d2 d ρ 2 2 + ρ(α1 + α2 − 1) + βρ ρ − m 2 f (ρ) = 0 dρ dρ 2 d d 2 + {(α1 − 1) tan φ + (α2 − 1) cot φ} − m g(φ) = 0 dφ 2 dφ 2 α3 − 1 d d 2 + β + z h(z) = 0 dz 2 z dz where, m is a constant (integer usually). In addition,

(4.3.13) (4.3.14) (4.3.15)

4.3 Cylindrical Wave Propagation in Fractional Space

45

βρ2 + βz2 = β 2

(4.3.16)

Equation (4.3.16) is referred to as constraint equation. In addition βρ , βz are known as wave constants in the ρ, z directions, respectively, which will be determined using boundary conditions. Now, Eq. (4.3.13) through (4.3.15) are needed to be solved for f (ρ), g(φ) and h(z), respectively. We choose to work first with f (ρ). Equation (4.3.13) can be written as: 2 d 2 d + aρ (4.3.17) + bρ + c f (ρ) = 0 ρ dρ 2 dρ where, a = α1 + α2 − 1, b = βρ2 , c = −m 2 , = 2. Equation (4.3.15) is closely related to Bessel’s equation and its solutions is given as [3]: 1−a 2√ 2√ 2 2 2 C1 Jv ( bρ ) + C2 Yv ( bρ ) (4.3.18) f (ρ) = ρ where, v = 1 (1 − a)2 − 4c Using (4.3.18), the final solution of (4.3.13) is given by f 1 (ρ) = ρ 1− or f 2 (ρ) = ρ 1−

α1 +α2 2

α1 +α2 2

C1 Jv (βρ ρ) + C2 Yv (βρ ρ)

D1 Hv(1) (βρ ρ) + D2 Hv(2) (βρ ρ)

(4.3.19)

(4.3.20)

where, v = 12 (2 − α1 − α2 )2 + 4m 2 . In (4.3.19) Jv (βρ ρ) is referred to as Bessel function of the first kind of order v and Yv (βr ) as the Bessel function of the second kind of order v. They are used to represent standing waves. In (4.3.20) Hv(1) (βr ) (2) is referred to as Hankel function of the first kind of order v and Hv (βρ ρ) as the Hankel function of the second kind of order v, and are used to represent traveling waves. Now, we find the solution of equation (4.3.14) for g(φ). Equation (4.3.14) can be reduced to following Gaussian hypergeometric equation after proper mathematical steps under substitution ξ = sin2 (φ) [3]: ξ(1 − ξ )

dg(φ) d 2 g(φ) + {(A + B + 1)ξ − C} + ABg(φ) = 0 2 dξ dξ

(4.3.21)

where, 1 (2 − α2 + α1 ) 2 m2 AB = − 4

A+ B+1=

(4.3.22) (4.3.23)

46

4 Electromagnetic Wave Propagation in Fractional Space

C=

1 (2 − α2 ) 2

(4.3.24)

solution to equation (4.3.21) is given as [3]: g(φ) = C3 F(A, B, C; ξ ) + C4 ξ 1−C F(A − C + 1, B − C + 1, 2 − C; ξ ) (4.3.25) where, F(A, B, C; ξ ) = 1 +

∞ (A)k (B)k ξ k (C)k k!

(4.3.26)

k=1

with, (A)k = A(A + 1) . . . (A + k + 1)

(4.3.27)

F(A, B, C; ξ ) is known as Gaussian hypergeometric function, and A, B, C are known from (4.3.22) through (4.3.24). Now, we find the solution of equation (4.3.15) for h(z). Equation (4.3.15) can be written as: 2 d d 2 (4.3.28) + βz z h(z) = 0 z 2 +e dz dz where, e = α3 − 1. Equation (4.3.28) is reducible to Bessel’s equation under substitution h = z n ζ as follows: d2 d |1 − e| z2 2 + z + (βz2 z 2 − n 2 ) ζ (z) = 0, n = (4.3.29) dz dz 2 The solution of Bessel’s equation in (4.3.29) is given as [3] ζ (z) = C5 Jn (βz z) + C 6 Yn (βz z)

(4.3.30)

where, Jn (βz z) is referred to as Bessel function of the first kind of order n, Yn (βz z) as the Bessel function of the second kind of order n. Finally the solution of (4.3.15) becomes α3 (4.3.31) h(z) = z n C5 Jn (βz z) + C6 Yn (βz z) , n = 1 − 2 The appropriate solution forms of f (ρ), g(φ) and h(z) depend upon the problem. From (4.3.12), (4.3.19), (4.3.25) and (4.3.31), a typical solution for ψ(r, θ, φ) to represent the fields within a cylindrical geometry may take the form ψ(ρ, φ, z) = [ρ 1−

α1 +α2 2

+ C4 ξ

{C 1 Jv (βρ ρ) + C2 Yv (βρ ρ)}] × [{C3 F(A, B, C; ξ )

F(A − C + 1, B − C + 1, 2 − C; ξ )}] × [z {C 5 Jn (βz z) + C 6 Yn (βz z)}] n

1−C

(4.3.32)

4.3 Cylindrical Wave Propagation in Fractional Space

47

where, ξ = sin2 (φ) and C1 through C6 are constant coefficients. Equation (4.3.32) provides a general solution to cylindrical wave equation in fractional space. This solution can be used to study the phenomenon of electromagnetic wave propagation in any non-integer dimensional space.

4.3.2 Discussion on Cylindrical Wave Solution in Fractional Space Equation (4.3.32) is the generalization of the concept of wave propagation from integer dimensional space to the non-integer dimensional space. As a special case, for three-dimensional space, this problem reduces to classical wave propagation concept; i.e., as a special case, if we set α1 = α2 = α3 = 1 in Eqs. (4.3.19), (4.3.25) and (4.3.31), we get cylindrical wave solution in integer dimensional space. For α1 = α2 = 1 Eqs. (4.3.19) and (4.3.20) provide f 1 (ρ) = C 1 Jm (βρ ρ) + C 2 Ym (βρ ρ)

(4.3.33)

f 2 (ρ) = D1 Hm(1) (βρ ρ) + D2 Hm(2) (βρ ρ)

(4.3.34)

and

Similarly, if we set α1 = α2 = 1 in Eqs. (4.3.22) and (4.3.24), we get A = −B = C = 21 . Now, considering following special forms of Gaussian hypergeometric function [4]: m 2,

1 F(λ, −λ, ; sin2 v) = cos(2λv) 2 sin[(2λ − 1)v] 3 F(λ, 1 − λ, ; sin2 v) = 2 (2λ − 1) sin(v)

(4.3.35) (4.3.36)

Eq. (4.3.25) can be reduced to g(φ) = C3 cos(mφ) + C4 sin(mφ) In a similar way, if we set α3 = 1 in (4.3.32) then n = 12 and it gives 1 h(z) = z 2 C 5 J 1 (βz z) + C6 Y 1 (βz z) 2

2

Using Bessel functions of fractional order [4]: 2 sin (z) J 1 (z) = 2 πz 2 cos (z) Y 1 (z) = − 2 πz

(4.3.37)

(4.3.38)

(4.3.39) (4.3.40)

48

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.6 Cylindrical waveguide of circular cross section.[This figure was originally published in [6] , reproduced courtesy of The Electromagnetics Academy]

equation (4.3.13) can be reduced to h(z) = C5 sin(βz z) + C6 cos(βz z)

(4.3.41)

where, Ci = Ci πβ2 z , i = 5, 6 From (4.3.12), (4.3.33), (4.3.37) and (4.3.41), a typical solution in three dimensional space ( a special case of fractional space) for ψ(ρ, φ, z) to represent the fields within a cylindrical geometry will take the form ψ(ρ, φ, z) = C1 Jm (βρ ρ) + C2 Ym (βρ ρ) × [C3 cos(mφ) + C4 sin(mφ)] × [C5 sin(βz z) + C6 cos(βz z)] (4.3.42) which is comparable to the cylindrical wave solutions of the wave equation in integer dimensional space obtained by Balanis [1]. As an example, the fields inside a circular waveguide filled with fractal media of dimension D can be obtained by assuming a D-dimensional fractional space inside the circular waveguide. Within such circular waveguide of radius a (see Fig. 4.6), standing waves are created in the radial(ρ) direction, periodic waves in the φ-direction, and traveling waves in the z-direction. For the fields to be finite at ρ = 0 where Y v(βρ ρ) possesses a singularity, (4.3.32) reduces to ψ1 (ρ, φ, z) =[ρ 1−

α1 +α2 2

+ C4 ξ

{C1 Jv (βρ ρ)}] × [{C3 F(A, B, C; ξ )

1−C

F(A − C + 1, B − C + 1, 2 − C; ξ )}]

× [z n {C5 Hn(2) (βz z) + C6 Hn(1) (βz z)}]

(4.3.43)

4.3 Cylindrical Wave Propagation in Fractional Space

49

Fig. 4.7 Cylindrical wave propagation in Euclidean space (D = 3). [This figure was originally published in [6] , reproduced courtesy of The Electromagnetics Academy]

Fig. 4.8 Cylindrical wave propagation in fractional space (D = 2.5). [ This figure was originally published in [6] , reproduced courtesy of The Electromagnetics Academy]

To represent the fields in the region outside the cylinder, where three dimensional space is assumed because there is no fractal media outside the cylinder, a typical solution for ψ(ρ, φ, z) would take the form

50

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.9 Cylindrical wave propagation in fractional space (D = 2.1). [ This figure was originally published in [6] , reproduced courtesy of The Electromagnetics Academy]

ψ2 (ρ, φ, z) = C2 Hm(2) (βρ ρ) × [C3 cos(mφ) + C4 sin(mφ)] × [C5 sin(βz z) + C 6 cos(βz z)]

(4.3.44)

In the region outside the cylinder, outward traveling waves are formed, in contract to standing waves inside the cylinder. In this way, the general cylindrical wave solution in fractional space can be used to study the wave propagation in the cylindrical geometries containing fractal media. Now, as another example we assume that a cylindrical wave exists in a fractional space due to some infinite line source. Since the source do not vary with z, the fields will not vary with z but will propagate away from the source in ρ-direction. Also for simplicity, we choose to visualize only the radial amplitude variations of scalar field ψ in fractional space which is given by (4.3.32) as: ψ(ρ) = Aρ 1−

α1 +α2 2

Hv(2) (βρ ρ)

(4.3.45)

Also, if we choose a single parameter for non-integer dimension D where 2 < D ≤ 3, i.e, we take α2 = α3 = 1 so D = α1 + 2. In this case (4.3.45) becomes ψ(ρ) = Aρ

3−D 2

Hv(2) (βρ ρ)

(4.3.46)

In (4.3.46), using asymptotic expansions of Hankel functions [4] for ρ → ∞, we see that the amplitude variations of field ψ are related with radial distance ρ as D

ψ(ρ) ∝ ρ 1− 2 From (4.3.47),

(4.3.47)

4.3 Cylindrical Wave Propagation in Fractional Space

for D = 3,ψ(ρ) ∝

51

√1 ρ

for D = 2.5, ψ(ρ) ∝ for D = 2.1, ψ(ρ) ∝

1 ρ 0.25 1 ρ 0.05

Assuming a time dependency e jwt , the radial amplitude variations of scalar field ψ are shown for different values of dimension D in Figs 4.7, 4.8, 4.9. It is seen that the amplitude of cylindrical wave propagating in higher dimensional space decays rapidly.

4.3.3 Summary An exact solution of cylindrical wave equation for electromagnetic field in D-dimensional fractional space is presented. The obtained exact solution of cylindrical wave equation is a generalization of classical integer-dimensional solution to a non-integer dimensional space. For all investigated cases when D is an integer dimension, the classical results are recovered. The investigated solution provides a basis for the application of the concept of fractional space to the wave propagation phenomenon in fractal media.

4.4 Spherical Wave Propagation in Fractional Space In this section we provide an exact solution of the spherical wave equation in D-dimensional fractional space which describes the phenomenon of electromagnetic wave propagation in fractional space. In Sect. 4.4.1, we investigate full analytical solution of spherical wave equation in D-dimensional fractional space, where a parameter D is used to describe the measure distribution of space. In Sect. 4.4.2, solution of wave equation in integer-dimensional space is justified from the results of previous section. Finally, in Sect. 4.4.3, results are summarized.

4.4.1 Spherical Wave Equation in D-dimensional Fractional Space The problems that exhibit spherical geometries are needed to be solved using spherical coordinates. As for the case of rectangular geometries, the electric and magnetic fields of spherical geometry boundary-value problem must satisfy corresponding spherical wave equation [1]. Let us assume that the space in which fields must be solved is fractional dimensional, source-free and lossless. For source-free and lossless media, the vector wave equations for the complex electric and magnetic field intensities are given by the Helmholtz’s equation as follows:

52

4 Electromagnetic Wave Propagation in Fractional Space 2 ∇D E + β2E = 0

(4.4.1)

2 H + β2H = 0 ∇D

(4.4.2)

where, β 2 = ω2 με. Time dependency e jwt has been suppressed throughout the 2 is the Laplacian operator in D-dimensional fractional space discussion. Here ∇ D and is defined in spherical coordinates as follows [2]: 2 ∂2 D−1 ∂ D−2 ∂ 1 ∂ 2 = 2+ + + 2 ∇D ∂r r ∂r r ∂θ 2 tan θ ∂θ 2 1 D−3 ∂ ∂ + 2 + r sin θ ∂φ 2 tan φ ∂φ (4.4.3) where, 2 < D ≤ 3. Once the solution to any one of Eq. (4.4.1) and (4.4.2) in fractional space is known, the solution to the other can be written by an interchange of E with H or H with E due to duality. We will examine the solution for E. In spherical coordinates, a general solution for E can be written as E(r, θ, φ) = aˆr Er (r, θ, φ) + aˆ θ E θ (r, θ, φ) + aˆ φ E φ (r, θ, φ)

(4.4.4)

Substituting (4.4.4) into (4.4.1) we can write that 2 (aˆ r E r + aˆ θ E θ + aˆ φ E φ ) + β 2 (aˆ r Er + aˆ θ E θ + aˆ φ E φ ) = 0 ∇D

(4.4.5)

Since, 2 2 (aˆ r Er ) = aˆ r ∇ D Er ∇D

(4.4.6)

2 2 ∇D (aˆ θ E θ ) = aˆ θ ∇ D Eθ

(4.4.7)

2 2 (aˆ φ E φ ) = aˆ φ ∇ D Eφ ∇D

(4.4.8)

So, Eq. (4.4.5) cannot be reduced to three simple scalar wave equations, but it can be reduced to three coupled scalar partial differential equations. However for simplicity, the wave mode solution can be formed in spherical coordinates that must satisfy the following scalar wave equation: 2 ψ(r, θ, φ) + β 2 ψ(r, θ, φ) = 0 ∇D

(4.4.9)

where, ψ(r, θ, φ) is a scalar function that can represent a field or vector potential component. In expanded form (4.4.9) can be written as 1 ∂ 2ψ D − 1 ∂ψ D − 2 ∂ψ ∂2ψ + 2 + + ∂r 2 r ∂r r ∂θ 2 tan θ ∂θ 2 1 D − 3 ∂ψ ∂ ψ + 2 + + β 2ψ = 0 r sin θ ∂φ 2 tan φ ∂φ (4.4.10)

4.4 Spherical Wave Propagation in Fractional Space

53

Equation (4.4.10) is separable using method of separation of variables. We consider ψ(r, θ, φ) = f (r )g(θ )h(φ)

(4.4.11)

the resulting ordinary differential equations are obtained as follows: 2 d 2 d 2 + (D − 1)r + (βr ) − n(n + 1) f = 0 r dr 2 dr

m 2 D−2 d d2 + + n(n + 1) g=0 − dθ 2 tan θ dθ sin θ 2 D−3 d d 2 + + m h=0 dφ 2 tan φ dφ

(4.4.12)

(4.4.13) (4.4.14)

where, m and n are constants (integers usually). Now, Eq. (4.4.12) through (4.4.14) are needed to be solved for f (r ), g(θ ) and h(φ), respectively. For simplicity we write f (r ), g(θ ), h(φ) as f, g, h, respectively, inside all equations. We choose to work first with f (r ). Equation (4.4.12) can be written as: 2 s d 2 d + ar (4.4.15) + br + c f = 0 r dr 2 dr where, a = D − 1, b = β 2 , c = −n(n + 1), s = 2. Equation (4.4.15) is closely related to Bessel’s equation and its solutions is given as [3]: 1−a 2√ s 2√ s f = r 2 C1 Jv ( br 2 ) + C2 Yv ( br 2 ) (4.4.16) s s where, v = 1s (1 − a)2 − 4c Using (4.4.16), the final solution of (4.4.12) is given by D

f 1 = r 1− 2 [C1 Jv (βr ) + C2 Yv (βr )]

(4.4.17)

D f 2 = r 1− 2 D1 Hv(1) (βr ) + D2 Hv(2) (βr )

(4.4.18)

or

where, v = 12 (2 − D)2 + 4n(n + 1). In (4.4.17) Jv (βr ) is referred to as Bessel function of the first kind of order v and Yv (βr ) as the Bessel function of the second kind of order v. They are used to represent radial standing waves. In (4.4.18) Hv(1) (βr ) (2) is referred to as Hankel function of the first kind of order v and Hv (βr ) as the Hankel function of the second kind of order v, and are used to represent radial traveling waves.

54

4 Electromagnetic Wave Propagation in Fractional Space

Now, we find the solution of Eq. (4.4.13) for g(θ ). Equation (4.4.13) can be reduced to following Gaussian hypergeometric equation after proper mathematical : steps under substitution g = w sinm θ and z 1 = 1+cosθ 2 z 1 (1 − z 1 )

d2w dw + {(α1 + β1 + 1)z 1 − γ1 } + α1 β1 w = 0 2 dz 1 dz 1

(4.4.19)

where, α1 + β1 + 1 = −2(D − 2)(n + 1)

(4.4.20)

α1 β1 = (n − m)(m + n − 1)

(4.4.21)

γ1 = −(D − 2)(n + 1)

(4.4.22)

solution to Eq. (4.4.19) is given as [3]: w = C3 F(α1 , β1 , γ1 ; z 1 ) 1−γ1

+ C4 z 1

F(α1 − γ1 + 1, β1 − γ1 + 1, 2 − γ1 ; z 1 )

(4.4.23)

where, F(α1 , β1 , γ1 ; z 1 ) = 1 +

∞ (α1 )k (β1 )k z 1k (γ1 )k k!

(4.4.24)

k=1

with, (α1 )k = α1 (α1 + 1)...(α1 + k + 1)

(4.4.25)

F(α, β, γ ; z) is known as Gaussian hypergeometric function, and α1 , β1 , γ1 are known from (4.4.20) through (4.4.22). From (4.4.19) through (4.4.23), the final solution of (4.4.13) is given by g = [C 3 F(α1 , β1 , γ1 ; z 1 ) 1−γ1

+ C4 z 1

F(α1 − γ1 + 1, β1 − γ1 + 1, 2 − γ1 ; z 1 )]sin m θ

(4.4.26)

Now, we find the solution of equation (4.4.14) for h(φ). Equation (4.4.14) can be written as: 2 d d +q h =0 (4.4.27) + pcotφ dφ 2 dφ

4.4 Spherical Wave Propagation in Fractional Space

55

where, p = D − 3, and q = m 2 . Equation (4.4.27) can be reduced to following Gaussian hypergeometric equation after proper mathematical steps under substitution : z 2 = 1+cosφ 2 z 2 (1 − z 2 )

d 2h dh + {(α2 + β2 + 1)z 2 − γ2 } + α2 β2 h = 0 2 dz 2 dz 2

(4.4.28)

where, α2 + β2 = D − 5

(4.4.29)

α2 β2 = m 2

(4.4.30)

γ2 =

1 (4 − D) 2

(4.4.31)

From (4.4.28) through (4.4.31), the final solution of (4.1.4) is given as h = C5 F(α2 , β2 , γ2 ; z 2 ) 1−γ2

+ C6 z 2

F(α2 − γ2 + 1, β2 − γ2 + 1, 2 − γ2 ; z 2 )

(4.4.32)

where, F is Gaussian hypergeometric function described in (4.2.4) through (4.2.5), and α2 , β2 , γ2 are known from (4.2.9) through (4.3.1). The appropriate solution forms of f, g and h depend upon the problem geometry. From (4.4.11), (4.1.7), (4.4.26) and (4.4.32), a typical solution for ψ(r, θ, φ) to represent the fields within a spherical geometry may take the form D

ψ(r, θ, φ) = [r 1− 2 {C 1 Jv (βr ) + C2 Yv (βr )}] × [{C3 F(α1 , β1 , γ1 ; z 1 ) 1−γ

+ C 4 z 1 1 F(α1 − γ1 + 1, β1 − γ1 + 1, 2 − γ1 ; z 1 )}sin m θ ] × [C 5 F(α2 , β2 , γ2 ; z 2 ) 1−γ2

+ C6 z 2

F(α2 − γ2 + 1, β2 − γ2 + 1, 2 − γ2 ; z 2 )]

(4.4.33) 1+cosφ , z = and C through C are constant coefficients. where, z 1 = 1+cosθ 2 1 6 2 2 Equation (4.4.33) provides a general solution to spherical wave equation in fractional space. This solution can be used to study the phenomenon of electromagnetic wave propagation in a non-integer dimensional space.

4.4.2 Discussion on Fractional Space Solution Equation (4.4.33) is the generalization of the concept of wave propagation in integer dimensional space to the wave propagation in non-integer dimensional space. As

56

4 Electromagnetic Wave Propagation in Fractional Space

a special case, for three-dimensional space, this problem reduces to classical wave propagation concept; i.e., as a special case, if we set D = 3 in Eq. (4.4.17) then v = n + 21 and it gives 1 f 1 = √ C1 Jn+ 1 (βr ) + C2 Yn+ 1 (βr ) 2 2 r

(4.4.34)

here, regular Bessel functions in (4.4.34) are related to spherical Bessel function by [4] π jn (βr ) = (4.4.35) J 1 (βr ) 2βr n+ 2 π Y 1 (βr ) yn (βr ) = (4.4.36) 2βr n+ 2 from (4.4.35) and (4.4.36) we can reduce (4.3.4) to 2β f1 = [C1 jn (βr ) + C2 yn (βr )] π similarly, for three dimensional space we reduce (4.1.8) to 1 (1) (2) f 2 = √ C 1 H 1 (βr ) + C2 H 1 (βr ) n+ 2 n+ 2 r

(4.4.37)

(4.4.38)

here, regular Hankel functions in (4.4.38) are related to spherical Hankel function by [4] π (1) (1) h n (βr ) = (4.4.39) H 1 (βr ) 2βr n+ 2 π (2) (4.4.40) h (2) H 1 (βr ) n (βr ) = 2βr n+ 2 from (4.4.39) and (4.4.40), we can reduce (4.3.8) to 2β (2) D1 h (1) f2 = n (βr ) + D2 h n (βr ) π

(4.4.41)

Similarly, we set D = 3 in (4.2.26) and solving (4.1.19) for this special case, we get d m Pn (ξ ) d m Q n (ξ ) g = C3 sin m θ + C4 (4.4.42) dξ m dξ m where, ξ = cos θ , Pn (ξ ) and Q n (ξ ) are referred to as Legendre functions of first and second kind and are related to associated Legendre functions Pnm (ξ ) and Q m n (ξ ) as [4]

4.4 Spherical Wave Propagation in Fractional Space

57 m

Pnm (ξ ) = (1 − ξ 2 ) 2 m

2 2 Qm n (ξ ) = (1 − ξ )

d m Pn (ξ ) dξ m

(4.4.43)

d m Q n (ξ ) dξ m

(4.4.44)

From (4.4.42) through (4.4.44), we get g in three-dimensional space as : g = C3 Pnm (cos θ ) + C4 Q m n (cos θ )

(4.4.45)

In a similar way, we set D = 3 in (4.3.2) and using special forms of Gaussian hypergeometric functions [4], we get h in three dimensional space as: h = C5 cos(mφ) + C 6 sin(mφ)

(4.4.46)

From (4.4.11), (4.4.37), (4.4.45) and (4.4.46), a typical solution in three dimensional space ( a special case of fractional space) for ψ(r, θ, φ) to represent the fields within a spherical geometry will take the form ψ(r, θ, φ) =

2β [C1 jn (βr ) + C2 yn (βr )] π × [C3 Pnm (cos θ ) + C4 Q m n (cos θ )] × [C5 cos(mφ) + C6 sin(mφ)]

(4.4.47)

which is comparable to the solution of spherical wave equation in integer dimensional space obtained by Balanis [1]. Now, as an example we assume that a spherical wave exists in a fractional space due to some point source. For simplicity, we choose to visualize only the radial amplitude variations of scalar field ψ in fractional space which is given by (4.1.8) as: ψ(r ) = Ar 1− 2 Hv(2) (βr ) D

(4.4.48)

where, 2 < D ≤ 3. In (4.4.48), using asymptotic expansions of Hankel functions [1] for r → ∞, we see that the amplitude variations of field ψ are related with radial distance r as ψ(r ) ∝ r From (4.4.49), for D = 3,ψ(r ) ∝ r1 1 for D = 2.5, ψ(r ) ∝ r 0.75 1 for D = 2.1,ψ(r ) ∝ r 0.55

1−D 2

(4.4.49)

58

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.10 Spherical wave propagation in Euclidean space (D = 3). [ This figure was originally published in [7] , reproduced courtesy of The BRILL]

Fig. 4.11 Spherical wave propagation in fractional space (D = 2.5). [ This figure was originally published in [7] , reproduced courtesy of The BRILL]

Assuming a time dependency e jwt , the radial amplitude variations of scalar field ψ are shown for different values of dimension D in Figs. 4.10, 4.11, 4.12. In Fig. 4.10 a spherical wave propagating in Euclidean space of dimension D = 3 is shown, where the amplitude decays by 1r . In Figs. 4.3, 4.4, 4.5 a spherical wave propagating

4.4 Spherical Wave Propagation in Fractional Space

59

Fig. 4.12 Spherical wave propagation in fractional space (D = 2.1). [ This figure was originally published in [7] , reproduced courtesy of The BRILL]

in fractional space of dimension D = 2.5 is shown, where the amplitude decays 1 by r 0.75 . In Fig. 4.12 a spherical wave propagating in fractional space of dimension 1 D = 2.1 is shown, where the amplitude decays by r 0.55 . It is seen that the amplitude of spherical wave propagating in higher dimensional space decays rapidly as compared to another spherical wave propagating in relatively lower dimensional space.

4.4.3 Summary An exact solution of the spherical wave equation is obtained in D-dimensional fractional space. The obtained fractional space solution provides a generalization of electromagnetic wave propagation phenomenon from integer space to fractional space. For integer values of dimension D, the classical results are recovered from fractional solution. The presented fractional space solution of the wave equation can be used to describe the phenomenon of wave propagation in any fractal media.

References 1. C.A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989) 2. C. Palmer, P.N. Stavrinou, Equations of motion in a noninteger-dimension space. J. Phys. A 37, 6987–7003 (2004) 3. A.D. Polyanin, V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. (CRC Press, Boca Raton, 2003)

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4 Electromagnetic Wave Propagation in Fractional Space

4. M. Abramowitz, I.A. Stegun, Handbook of mathematical functions with formulas, Graphs, and Mathematical Tables.U.S. Department of Commerce (1972) 5. M. Zubair, M.J. Mughal, Q.A. Naqvi, The wave equation and general plane wave solutions in fractional space. Prog Electromn Res Lett 19, 137–146 (2010) 6. M. Zubair, M.J. Mughal, Q.A. Naqvi, An exact solution of cylindrical wave equation for electromagnetic field in fractional dimensional space. Prog Electromn Res Lett 114, 443–455 (2011) 7. M. Zubair, M.J. Mughal, Q.A. Naqvi, An exact solution of spherical wave in D-dimensional fractional space. J Electromn Waves Appl 25, 1481–1491 (2011)

Chapter 5

Electromagnetic Radiations from Sources in Fractional Space

In this chapter we present a procedure for solution of antenna radiation problems in fractional space along with an application of this novel procedure to the case of Hertzian dipole in fractional space. In Sect. 5.1, a novel solution procedure for antenna radiation problems in fractional space is proposed. In Sect. 5.2, the reported solution procedure is applied to the case of Hertzian dipole in fractional space. And finally major results are summarized in Sect. 5.3

5.1 Solution Procedure for Radiation Problems in Fractional Space In analysis of radiation problems, the procedure is to specify sources and get the fields radiated by the sources. A common practice in analysis of radiation problems in fractional space is to introduce auxiliary potential function AD (magnetic vector potential) and FD (electric vector potential) [1]. An overview of steps involved in solving typical radiation problems in fractional space are shown in Fig. 5.1 .

5.1.1 The Vector Potential AD for Electric Current Source J The vector potential AD in D-dimensional fractional space is useful in solving for the electromagnetic field generated by a given harmonic electric current J [1]. The fractional space generalization of the relation between AD and J is given by vector potential wave equation as below: 2 ∇D AD + k 2 AD = −μJ

(5.1.1)

2 is the laplacian operator in D-dimensional fractional space given by where, ∇ D Eq. 3.2.3, and k 2 = ω2 με.

M. Zubair et al., Electromagnetic Fields and Waves in Fractional Dimensional Space, SpringerBriefs in Applied Sciences and Technology, DOI: 10.1007/978-3-642-25358-4_5, © The Author(s) 2012

61

62

5 Electromagnetic Radiations from Sources in Fractional Space

Fig. 5.1 Block diagram for computing radiated fields in fractional space

Using the solution of Poisson’s equation in fractional space [2] and considering analogy of (5.1.1) with vector wave equation in fractional space solved in Sect. 4.1, we solve (5.1.1) for AD and get AD =

μ 23−D Γ (3/2) 4π Γ (D/2)

J(x , y , z )

(2)

Hn (k R) dV R D−5/2

(5.1.2)

where, the primed coordinates (x , y , z ) represent the location of source, n = 1− D2 , R is the distance between any point in the source to observation point, Γ (x) is the (2) gamma function and Hn (k R) denotes the Hankel function of second kind of order n representing outward going waves from source point. Now, to validate our provided solution in (5.1.2), we get vector potential A from our solution by substituting D = 3 in (5.1.2). For D = 3, we have n = −1/2. Using following Hankel function of fractional order [3] 2 −jx 2 (5.1.3) e H 1 (x) = πx 2 Equation 5.1.2 gets reduced to AD = C

μ 4π

J(x , y , z )

e− jk R dV R

(5.1.4)

where, C is a constant term. Equation 5.1.4 is in exact agreement with the solution provided in [1] for Euclidean space.

5.1.2 The Vector Potential FD for Magnetic Current Source M Although magnetic currents appear to be physically unrealizable, equivalent magnetic currents are considered when we use surface or volume equivalence theorems [4]. Similar to previous case, the vector potential FD in D-dimensional fractional space is useful in solving for the electromagnetic field generated by a given harmonic electric current H [1]. The fractional space generalization of the relation between FD and M is given by vector potential wave equation as follows: 2 FD + k 2 FD = −εM ∇D

(5.1.5)

5.1 Solution Procedure for Radiation Problems in Fractional Space

63

2 is the laplacian operator in D-dimensional fractional space given by where, ∇ D (3.2.3), and k 2 = ω2 με. Using the analogy to the solution provided for AD in (5.1.2), we get

ε 23−D Γ (3/2) FD = 4π Γ (D/2)

M(x , y , z )

(2)

Hn (k R) dV R D−5/2

(5.1.6)

where, the primed coordinates (x , y , z ) represent the location of source, n = 1− D2 , R is the distance between any point in the source to observation point, Γ (x) is the (2) gamma function and Hn (k R) denotes the Hankel function of second kind or order n representing outward going waves from source point. Now, to validate our provided solution in (5.1.6), we get vector potential F from our solution by substituting D = 3 in (5.1.6). For D = 3, we have n = −1/2. Using Hankel function of fractional order in (5.1.3), finally Eq. 5.1.6 gets reduced to ε e− jk R (5.1.7) FD = C dV M(x , y , z ) 4π R Equation 5.1.7 is in exact agreement with the solution provided in [1] for Euclidean space.

5.1.3 Radiated Electric and Magnetic Fields in Far Zone for Electric J and Magnetic Current Source M In the previous Sections, we have developed equations that can be used for electric and magnetic fields generated by and electric current source J and a magnetic current source M in fractional space. The procedure for radiation analysis in fractional space requires that the potential functions AD and FD are generated, respectively, by J and M. In turn the corresponding electric and magnetic fields are then determined in far zone. The fields radiated in far zone (k R >> 1) due to AD are EAD and HAD and are given by [1] EAD = − jωAD HAD =

− jω aˆ r × AD η

(5.1.8) (5.1.9)

where η is wave impedance. The fields radiated due to FD are EFD and HFD and are given in the same form as (5.1.8) and (5.1.9): HFD = − jωFD EFD =

− jω aˆ r × FD η

(5.1.10) (5.1.11)

64

5 Electromagnetic Radiations from Sources in Fractional Space

And finally, the total fields are given by superposition of the individual fields due to AD and FD as: ED = EAD + EFD

(5.1.12)

HD = HAD + HFD

(5.1.13)

5.2 Elementary Hertzian Dipole in Fractional Space Consider a Hertzian dipole ( an infinitesimal linear wire with l

For further volumes: http://www.springer.com/series/8884

Muhammad Zubair Muhammad Junaid Mughal Qaisar Abbas Naqvi •

Electromagnetic Fields and Waves in Fractional Dimensional Space

123

Muhammad Zubair Faculty of Electronic Engineering GIK Institute of Engineering Sciences and Technology Topi Pakistan e-mail: [email protected]

Qaisar Abbas Naqvi Department of Electronics Quaid-e-Azam University Islamabad Pakistan e-mail: [email protected]

Muhammad Junaid Mughal Faculty of Electronic Engineering GIK Institute of Engineering Sciences and Technology Topi Pakistan e-mail: [email protected]

ISSN 2191-530X ISBN 978-3-642-25357-7 DOI 10.1007/978-3-642-25358-4

e-ISSN 2191-5318 e-ISBN 978-3-642-25358-4

Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011942412 Ó The Author(s) 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

It’s ironic that fractals, many of which were invented as examples of pathological behavior, turn out to be pathological at all. In fact they are the rule in the universe. Shapes, which are not fractal, are the exception. I love Euclidean geometry, but it is quite clear that it does not give a reasonable presentation of the world. Mountains are not cones, clouds are not spheres, trees are not cylinders, neither does lightning travel in a straight line. Almost everything around us is non-Euclidean. Benoit Mandelbrot, 1924

To my beloved father Mr. Hafiz Muhammad Makhdoom whose utmost efforts since my childhood make me what I am today M. Zubair To my father Mr. Abdul Ghafoor Mughal for his love and kindness when he was alive and his beautiful memories when he is no longer with us M. J. Mughal To my parents Q. A. Naqvi

Preface

The concept of fractional dimensional space is being effectively used in many areas of physics to describe the effective parameters of physical systems. Although the space, embedding things, in real world is three dimensional Euclidean space, the material objects are not always moving in three dimensional space. The dimensionality depends upon the restraint conditions. The phenomenon of electromagnetic wave propagation, radiation and scattering in fractal structures can be described by replacing these confining fractal structures with a D-dimensional fractional space. Thus, given this simple value of D, the real system can be modeled in a simple analytical way. With this view, a theoretical investigation of electromagnetic fields and waves in fractional dimensional space is provided in this book which is useful to study the behavior of electromagnetic fields and waves in fractal media. A novel fractional space generalization of the differential electromagnetic equations is provided. A new form of vector differential operators is formulated in fractional space. Using these modified vector differential operators, the classical Maxwell’s electromagnetic equations have been worked out. The Laplace’s, Poisson’s and Helmholtz’s equations in fractional space are derived by using modified vector differential operators. A fractional space generalization of potentials for static and time-varying fields is presented by solving Laplace’s equation and inhomogeneous vector wave equation, respectively, in fractional space. The phenomenon of electromagnetic wave propagation in fractional space is studied in detail by providing full analytical plane-, cylindrical- and spherical-wave solutions of the vector wave equation in D-dimensional fractional space. An analytical solution procedure for radiation problems in fractional space has also been proposed. As an application, the fields radiated by a Hertzian dipole in fractional space have been worked out. For all the investigated cases when integer dimensional space is considered, the classical results are recovered. The differential electromagnetic equations in fractional space, established in this book, provide a basis for application of the concept of fractional space in solving electromagnetic wave propagation, radiation and scattering problems in fractal media.

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This book has been divided into six chapters. In Chap. 2, a novel generalization of differential electromagnetic equations in fractional space is provided on the basis of modified vector differential operators for fractional space. A new form of vector differential operator Del, written as rD, and its related differential operators is formulated in fractional space. Using these modified vector differential operators, the classical Maxwell’s electromagnetic equations have been worked out. The Laplace’s, Poisson’s and Helmholtz’s equations in fractional space are also derived by using modified vector differential operators. Also a new fractional space 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 established differential electromagnetic equations in fractional space. In Chap. 3, a fractional space generalization of potentials for static and timevarying fields is presented by solving Laplace’s equation and inhomogeneous vector wave equation, respectively, in fractional space. In Chap. 4, the phenomenon of wave propagation in fractional space is investigated by solving Helmholtz’s equation in different coordinate systems. General plane wave solutions, in source-free and lossless as well as lossy media, in fractional space are presented by solving vector wave equation in D-dimensional fractional space. An exact solution of cylindrical as well as spherical wave equation, for electromagnetic field in D-dimensional fractional space, is also presented. All these investigated solutions of vector wave equation provide a basis for the application of the concept of fractional space to the wave propagation phenomenon in fractal media. For all investigated cases when integer dimension is considered, the classical results were recovered to validate obtained results. Chapter 5 deals with the solution procedure for radiation problems in fractional space.The proposed solution procedure can be used to study the radiation phenomenon in any non-integer dimensional fractal media. As an application, the fields 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 emerging area of research in the field of electromagnetics. The concept of fractional dimensional space has potential to make a significant impact on future directions in fractional electromagnetics research. We highly recommend this book to graduate students, researchers, and professionals working in the areas of electromagnetic-wave propagation, radiation, scattering, diffraction, and other related fields of applied mathematics. The topics in this book can also be covered in any graduate course on ’’Advanced Engineering Electromagnetics’’. Pakistan September 2011

Muhammad Zubair Muhammad Junaid Mughal Qaisar Abbas Naqvi

Acknowledgments

This book is an enlarged form of Authors’ work on fractional dimensional space electromagnetics published in different journals. Some figures from published work have been reproduced with prior permission and are cited with full acknowledgement to corresponding source. We would like to sincerely thank the GIK Institute of Engineering Sciences and Technology, Topi, Pakistan, for providing the necessary facilities to accomplish this work. We would also take this opportunity to thank all our friends and colleagues 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 is the person who taught us the subject of electromagnetics and nurtured our interest in this field. We feel extremely fortunate to have learnt this subject from him and are sure to say that this work could not have been accomplished without his guidance. Finally, we are very thankful to Dr. Christoph Baumann and Mrs. CarmenWolf at Springer-Verlag GmbH for their wonderful help in the preparation and publication of this manuscript. Pakistan September 2011

Muhammad Zubair Muhammad Junaid Mughal Qaisar Abbas Naqvi

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fractional Dimensional Space . . . . . . . . . . . . . . . . . . . 1.2 Axiomatic Basis for Fractional Dimensional Space . . . . 1.3 Differential Geometry of Fractional Dimensional Space . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Differential Electromagnetic Equations in Fractional Space . . . . . 2.1 Fractional Space Generalization of Laplacian Operator . . . . . . 2.2 Fractional Space Generalization of Del Operator and Related Differential Operators. . . . . . . . . . . . . . . . . . . . . 2.2.1 Del Operator in Fractional Space . . . . . . . . . . . . . . . . 2.2.2 Gradient Operator in Fractional Space . . . . . . . . . . . . . 2.2.3 Divergence Operator in Fractional Space . . . . . . . . . . . 2.2.4 Curl Operator in Fractional Space . . . . . . . . . . . . . . . . 2.3 Fractional Space Generalization of Differential Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fractional Space Generalization of Potentials for Static Fields, Poisson’s and Laplace’s Equations. . . . . . . . . . . . . . . . . . . . . 2.5 Fractional Space Generalization of Potentials for Time-Varying Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Fractional Space Generalization of the Helmholtz’s Equation . . 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Potentials for Static and Time-Varying Fields in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Electrostatic Potential in Fractional Space . . . . . . . . . . . . . . . . 3.1.1 An Exact Solution of the Laplace’s Equation in D-dimensional Fractional Space . . . . . . . . . . . . . . . .

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3.1.2

Electrostatic Potential Inside a Rectangular Box in Fractional Space . . . . . . . . . . . . . . . . . . . . . 3.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Time-Varying Potentials in Fractional Space. . . . . . . . . 3.2.1 Inhomogeneous Vector Potential Wave Equation in D-dimensional Fractional Space . . . . . . . . . . 3.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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Electromagnetic Wave Propagation in Fractional Space . . . . 4.1 General Plane Wave Solutions in Fractional Space: Lossless Medium Case . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 General Plane Wave Solutions in Fractional Space . 4.1.2 Discussion on Fractional Space Solution . . . . . . . . 4.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 General Plane Wave Solutions in Fractional Space: Lossy Medium Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 General Plane Wave Solutions in Lossy Medium in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Discussion on Fractional Space Solution in Lossy Medium . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Example: Current Sheet as Source of Plane Waves in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Cylindrical Wave Propagation in Fractional Space . . . . . . 4.3.1 An Exact Solution of Cylindrical Wave Equation in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Discussion on Cylindrical Wave Solution in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Spherical Wave Propagation in Fractional Space . . . . . . . . 4.4.1 Spherical Wave Equation in D-dimensional Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Discussion on Fractional Space Solution . . . . . . . . 4.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Electromagnetic Radiations from Sources in Fractional Space . . 5.1 Solution Procedure for Radiation Problems in Fractional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Vector Potential AD for Electric Current Source J 5.1.2 The Vector Potential FD for Magnetic Current Source M . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.1.3

Radiated Electric and Magnetic Fields in Far Zone for Electric J and Magnetic Current Source M . . . . 5.2 Elementary Hertzian Dipole in Fractional Space . . . . . . . . 5.2.1 Fields Radiated . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Directivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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About the Authors

Mr. Muhammad Zubair did his BS in Electronic Engineering with Gold Medal from International Islamic University, Islamabad, Pakistan in 2009. Recently, he has completed his MS in Electronic Engineering with Highest Distinction from GIK Institute of Engineering Sciences and Technology, Topi, Pakistan in 2011 and joined the same institute as Research Associate. Mr. Zubair’s research interests are in the field of Analytical Electromagnetics. He has applied the concept of fractional dimensional space in the study of electromagnetic wave propagation, radiation and scattering in fractal media. He is also member of Pakistan Engineering Council. Dr. M. Junaid Mughal did his M.Sc and M.Phil in Electronics from Quaid-eAzam university, Islamabad in 1993 and 1995, respectively. He did his PhD from the University of Birmingham, UK in 2001. He worked as Director of Engineering in Nuonics Inc., Orlando, Fl, USA form 2001 to 2003. He is presently working as Associate Professor in the Faculty of Electronic Engineering in GIK Institute. Dr. Mughal’s research interests are primarily in the field of communications and particularly in RF and Optical Communications. He has worked in antennas, EM scattering, propagation modeling for mobile applications and fiber optics. In the field of optical communication Dr Mughal is coinventor of high dynamic range variable optical attenuators based on Acousto-Optic and MEMS technology, high speed fiber-optic switches, fiber optic tunable filters and laser beam profiling systems. Currently he is working in the area of tunable metamaterials, wave propagation in fractal media and focusing systems embedded in Chiral medium. Dr. Qaisar Abbas Naqvi completed his M.Sc., M.Phil., and Ph.D., all in Electronics, from Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan in 1991, 1993, and 1998 respectively. He joined Department of Electronics as Assistant Professor in 1998. Uptill now, he has successfully supervised more than thirty M.Phil and nine PhD students. He is now Associate Professor 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 more than 10 international journals. His research interests include fractional paradigm in electromagnetics, bi-isotropic and chiral mediums, high frequency techniques and Kobayashi potential method.

Chapter 1

Introduction

This book is a theoretical investigation of electromagnetic fields and waves in the fractional dimensional space. The motivation for this study, besides its theoretical importance, is provided by its applicability to the problems of electromagnetic wave modeling in complex fractal media. One of the important advantages of fractals is their capability to model objects of complicated structures. This is because of an important property of fractals that their structure is characterized by a small number of parameters. One of those parameters is the fractional dimension which tells how the fractal fills the Euclidean space in which it lies. Since, a medium composed of such fractal objects can be considered as non-integer dimensional fractal media, the analytical results of this work provide the necessary tools for analyzing the behavior of electromagnetic fields and waves in it.

1.1 Fractional Dimensional Space Every one of us has learnt that the lines and curves are one-dimensional, planes and surfaces are two-dimensional, solids and volumes are three dimensional, and so on. In a formal way, we say that a set is n-dimensional if we have n independent variables to describe a neighborhood of any point. Such a notion of dimension is called the “topological dimension” of a set. Also we observe that if we take the union of infinite many sets of n dimension, the overall dimension of new set can grow to n+1 e.g., the union of infinite number of (one-dimensional) lines give rise to a (two-dimensional) plane. Now, we think about the dimension in another way. We may break a line segment into 4 self-similar intervals, each with the same length, and each of which can be magnified by a factor of 4 to yield the original segment. We can also break a line segment into 7 self-similar pieces, each with magnification factor 7 can yield the original segment, or 20 self-similar pieces with magnification factor 20 to yield the original segment. In general, we can break a line segment into N self-similar pieces, each with magnification factor N to yield the original segment. If, we decompose a

M. Zubair et al., Electromagnetic Fields and Waves in Fractional Dimensional Space, SpringerBriefs in Applied Sciences and Technology, DOI: 10.1007/978-3-642-25358-4_1, © The Author(s) 2012

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1 Introduction

square into 4 self-similar sub-squares, and the magnification factor here will be 2 to yield in the original square. Alternatively, we can break the square into 9 self-similar pieces with magnification factor 3, or 25 self-similar pieces with magnification factor 5. Clearly, the square may be broken into N 2 self-similar copies of itself, each of which must be magnified by a factor of N to yield the original figure. Finally, we can decompose a cube into N 3 self-similar pieces, each of which has magnification factor N. Following above discussion we can say that the dimension is simply the exponent of the number of self-similar pieces with magnification factor N into which the figure may be broken. So what is the dimension of the Sierpinski triangle? How do we find the exponent in this case? For this, we need logarithms. Note that, for the square, we have N 2 self-similar pieces, each with magnification factor N. So we can write dimension =

log N 2 log(number of self similar pieces) = =2 log(magnification factor) log N

Similarly, the dimension of a cube is: dimension =

log N 3 log(number of self similar pieces) = =3 log(magnification factor) log N

Thus, we take as the definition of the fractal dimension of a self-similar object: dimension =

log(number of self similar pieces) log(magnification factor)

As the Sierpinski triangle consists of 3 self-similar pieces, each with magnification factor 2, So the fractal dimension is dimension =

log(number of self similar pieces) log 3 = ≈ 1.58 log(magnification factor) log 2

In general, the Sierpinski triangle breaks into 3N self-similar pieces with magnification factors 2N , so we again have fractal dimension =

log 3N log(number of self similar pieces) = ≈ 1.58 log(magnification factor) log 2N

This estimates that the Sierpinski triangle sits somewhere in between lines and planes. Similarly, many fractal structures are known in literature that possess a fractal dimension. Roughly speaking, we state that the space embedding such fractal curves or surfaces is known as “fractional dimensional space”. There has been much interest to study different physical phenomenon in fractional dimensional space [1–28] during the last few decades. The concept of fractional space is used to replace the real anisotropic confining structure with an isotropic fractional

1.1 Fractional Dimensional Space

3

space, where the measurement of this confinement is given by fractional dimension [6, 7]. It is also important to mention that the experimental measurement of the dimension of real world is 3 ± 10−6 , not exactly 3 [6, 9]. Among several methods, a methodology to describe the fractional dimension is fractional calculus [29], which is also used by different authors [30–37] in studying various electromagnetic problems. Axiomatic basis for spaces with fractional dimension have been provided by Stillinger [6], along with a fractional space generalization of Laplacian operator and a solution of Schrödinger’s wave equation in fractional dimensional space. For 2-spatial coordinate space, the Stillinger’s formalism shows that it is possible to distribute the D dimensions between them. Palmer and Stavrinou [8] generalized the results of Stillinger to n orthogonal coordinates. Equations of motion in a non-integer dimensional space have also been formulated in [8]. The formalism investigated in [8] allows to describe an anisotropic confinement of fractional space, i.e., if we have a system that is confined as 1.8 dimensional, then it could be described as 1 + 0.8 dimensional in two coordinates and as 1 + 0.2 + 0.6 dimensional in three coordinates. Recently, Muslih [18] provided a dimensional regularization technique in order to convert any integral of a function from fractional dimensional space to a regular dimensional space along with a description of differential geometry of fractional dimensional space. The generalization of electromagnetic theory in fractional space is of much importance to study the phenomenon of wave propagation, radiation and scattering in an anisotropic fractal media. Fractal models of media are becoming popular due to relatively small number of parameters that define a medium of greater complexity and rich structure [15]. In general, the fractal media cannot be considered as continuous media, because some of points and domains are not filled by the medium particles. These unfilled domains are called porous. The fractal media can be treated as continuous media for the scales much larger than average pore size. In order to describe the fractal media, the continuous medium model for fractal media reported in [16], suggests to use the space with fractional dimension. An introductory work on fractional multipoles and electromagnetic field in fractional space is reported in [17–20]. It is worthwhile to mention that clouds, turbulence in fluids, rough surfaces, snow, etc., can be described as fractional dimensional. The study of wave propagation, radiation and scattering phenomenon in such media is important in practical applications, such as communications, remote sensing, navigation and even bioengineering [20].

1.2 Axiomatic Basis for Fractional Dimensional Space This work is based on the Stillinger’s [6] axiomatic basis for spaces with non-integer dimension. Here, we briefly describe these axioms. In [6], four topological axioms are proposed which generate a space with non-integer dimension D. Let SD denote the fractional space which contains points x, y, . . . and has topological structure specified by the following axioms: Axiom 1 SD is a metric space.

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1 Introduction

Axiom 2 SD is dense in itself. Axiom 3 SD is metrically unbounded. Axiom 4 For any two points y, z ∈ SD , and any ∈ >0, there exists an x ∈ SD such that: (a) (b)

r(x, y) + r(x, z) = r(y, z) |r(x, y) − r(x, z)| < εr(y, z)

The full implication of Axiom 4 is that any two points in SD are connected by a continuous line embedded in that space so SD is connected. So any convex or star domain in SD will be contractible. Based on these axioms Stillinger [6] as well as recently Muslih [18], provided a dimensional regularization technique in order to convert any integral of a function from fractional dimensional space to a regular dimensional space. The fractional space generalization of the Laplacian operator, provided in later chapters, is based on the same dimensional regularization technique according to which a fractional space is related to fractional integrals and derivatives.

1.3 Differential Geometry of Fractional Dimensional Space The question of differential geometry of fractional spaces is related to the dimensional regularization technique. A sufficient description of differential geometry of fractional space along with dimensional regularization technique is already provided in [18] which is briefly described here: Let us say that N coordinates x1 , x2 , . . . , xN are needed to locate a point in a space. In the case where a space is filled with regular geometric objects, and the curves and the surfaces are smooth, it is common to call this number N as the dimension of the space. Thus, a straight line, a plane surface, and a cube are of dimensions 1, 2, and 3, respectively. This is also true if these spaces have curvatures. For example, motions along the circumference of a circle and on the surface of a sphere can be considered as motions in one- and two-dimensional spaces even though our true motions may be in a three-dimensional space. In such cases, infinitesimal line, area, and volume elements in the Cartesian coordinates are defined as dx1 , dx1 dx2 , and dx1 dx2 dx3 , respectively, and even in the case of a space with curvature, the distance between two points sufficiently closed to each other is given by a quadratic expression. However, this is not the situation in the case of fractal lines, surfaces, volume and hypervolumes. Thus, in these cases, there is a clear distinction between the number of coordinates used to locate a point and the dimension of the space. The dimensions of the fractional space can be defined in various ways. In [18], a scaling method d α x = f (α)|x|α−1 dx is used to relate the differential fractional line element d α x with dx, where 0 < α ≤ 1 is the fractional dimension of the line and α/2 f (α) = Γπ(α/2) is a function of α. A space filled with such lines is called fractional space. Thus, if a point in a space is located using N points x1 , x2 , . . . , xN , then the

1.3 Differential Geometry of Fractional Dimensional Space

5

scaling between d αi xi and dxi is defined as d αi xi =

π αi /2 |x|αi −1 dxi , i = 1, 2, . . . , N Γ (αi /2)

This space is called fractional dimensional space or simply a fractional space, and D = α1 + α2 +, . . . , +αN is the dimension of the fractional space. Following the above discussion, any volume element in D-dimensional fractional space can be defined as: d(V )D = d α1 x1 , d α2 x1 , . . . , d αN xN where, d αN xN is the differential element corresponding to Nth coordinate. In later chapters, we have established the differential electromagnetic equations in fractional dimension space to provide a basis for application of the concept of fractional dimensional space in solving electromagnetic wave propagation, radiation and scattering problems in fractal media.

References 1. M. Zubair, M.J. Mughal, Q.A. Naqvi, The wave equation and general plane wave solutions in fractional space. Prog. Electromagnet. Res. Lett. 19, 137–146 (2010) 2. M. Zubair, M.J. Mughal, Q.A. Naqvi, On electromagnetic wave propagation in fractional space. Non-linear Anal. B: Real World App. 12(5), 2844–2850 (2011) 3. M. Zubair, M.J. Mughal, Q.A. Naqvi, A.A. Rizvi, Differential electromagnetic equations in fractional space. Prog. Electromagnet. Res. 114, 255–269 (2011) 4. M. Zubair, M.J. Mughal, Q.A. Naqvi, An exact solution of cylindrical wave equation for electromagnetic field in fractional dimensional space. Prog. Electromagnet. Res. 114, 443–455 (2011) 5. M. Zubair, M.J. Mughal, Q.A. Naqvi, An exact solution of spherical wave in D-dimensional fractional space. J. Electromagn. Waves App. 25, 1481–1491 (2011) 6. F.H. Stillinger, Axiomatic basis for spaces with noninteger dimension. J. Math. Phys. 18(6), 1224–1234 (1977) 7. X. He, Anisotropy and isotropy: a model of fraction-dimensional space. PSolid State Commun. 75, 111–114 (1990) 8. C. Palmer, P.N. Stavrinou, Equations of motion in a noninteger-dimension space. J. Phys. A 37, 6987–7003 (2004) 9. K.G. Willson, Quantum field-theory, models in less than 4 dimensions. Phys. Rev. D 7(10), 2911–2926 (1973) 10. B. Mandelbrot, The Fractal Geometry of Nature. (W.H. Freeman, New York, 1983) 11. C.G. Bollini, J.J. Giambiagi, Dimensional renormalization: The number of dimensions as a regularizing parameter. Nuovo Cimento B 12, 20–26 (1972) 12. J.F. Ashmore, On renormalization and complex space-time dimensions. Commun. Math. Phys. 29, 177–187 (1973) 13. O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 271(1), 368–379 (2002) 14. D. Baleanu, S. Muslih, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scripta 72(23), 119–121 (2005)

6

1 Introduction

15. V.E. Tarasov, Electromagnetic fields on fractals. Modern Phys. Lett. A 21(20), 1587–1600 (2006) 16. V.E. Tarasov, Continuous medium model for fractal media. Phys. Lett. A 336(2–3), 167–174 (2005) 17. S. Muslih, D. Baleanu, Fractional multipoles in fractional space. Nonlinear Anal: Real World App. 8, 198–203 (2007) 18. S.I. Muslih, O.P. Agrawal, A scaling method and its applications to problems in fractional dimensional space. J. Math. Phys. 50(12):123501–123511 (2009) 19. D. Baleanu, A.K. Golmankhaneh, A.K. Golmankhaneh, On electromagnetic field in fractional space. Nonlinear Anal: Real World App. 11(1):288–292 (2010) 20. Z. Wang, B. Lu, The scattering of electromagnetic waves in fractal media. Waves Random Complex Media 4(1), 97–103 (1994) 21. C.M. Bender, K.A. Milton, Scalar casimir effect for a D-dimensional sphere. Phys. Rev. D 50, 6547–6555 (1994) 22. S. Muslih, D. Baleanu, Mandelbrot scaling and parametrization invariant theories. Romanian Rep. Phys. 62(4), 689–696 (2010) 23. S. Muslih, M. Saddallah, D. Baleanu, E. Rabe, Lagrangian formulation of maxwell’s field in fractional D dimensional space-time. Romanian Rep. Phys. 55(7–8), 659–663 (2010) 24. S. Muslih, O.P. Agrawal, Riesz fractional derivatives and fractional dimensional space. Int. J. Theor. Phys. 49(2):270–275 (2010) 25. S. Muslih, Solutions of a particle with fractional [delta]-potential in a fractional dimensional space. Int. J. Theor. Phys. 49(9), 2095–2104 (2010) 26. E. Rajeh, S.I. Muslih, B. Dumitru, E. Rabei, On fractional Schrodinger equation in [alpha]dimensional fractional space. Nonlinear Anal.: Real World App. 10(3):1299–1304 (2009) 27. M. Sadallah, S.I. Muslih, Solution of the equations of motion for einsteins field in fractional D dimensional space-time. Int. J. Theor. Phys. 48(12):3312–3318 (2009) 28. S. Muslih, D. Baleanu, E.M. Rabe, Solutions of massless conformal scalar field in an n-dimensional einstein space. Acta Phys. Pol. Ser. B 39(4):887–892 (2008) 29. K.B. Oldham, J. Spanier, The Fractional Calculus. (Academic Press, New York, 1974) 30. A. Hussain, Q.A. Naqvi, Fractional rectangular impedance waveguide. Prog. Electromagnet. Res. 96, 101–116 (2009) 31. Q.A. Naqvi, Planar slab of chiral nihility metamaterial backed by fractional dual/PEMC interface. Prog. Electromagnet. Res. 85, 381–391 (2008) 32. Q.A. Naqvi, Fractional dual interface in chiral nihility medium. Prog. Electromagnet. Res. Lett. 8, 135–142 (2009) 33. Q.A. Naqvi, Fractional dual solutions in grounded chiral nihility slab and their effect on outside fields. J. Electromagn. Waves App. 23(5–6), 773–784(12) (2009) 34. A. Naqvi, S. Ahmed, Q.A. Naqvi, Perfect electromagnetic conductor and fractional dual interface placed in a chiral nihility medium. J. Electromagn. Waves App. 24(14–15), 1991–1999(9) (2010) 35. A. Naqvi, A. Hussain, Q.A. Naqvi, Waves in fractional dual planar waveguides containing chiral nihility metamaterial. J. Electromagn. Waves App. 24(11–12), 1575–1586(12) (2010) 36. E.I. Veliev, M.V. Ivakhnychenko, T.M. Ahmedov, Fractional boundary conditions in plane waves diffraction on a strip. Prog. Electromagn. Res. 79, 443–462 (2008) 37. S.A. Naqvi, M. Faryad, Q.A. Naqvi, M. Abbas, Fractional duality in homogeneous bi-isotropic medium. Prog. Electromagn. Res. 78, 159–172 (2008)

Chapter 2

Differential Electromagnetic Equations in Fractional Space

In this chapter a novel generalization of differential electromagnetic equations in fractional space is provided. Firstly, basic vector differential operators are generalized in fractional space and then using these fractional operators Maxwell’s, Laplace’s, Poisson’s and Helmholtz’s equations have been worked out in fractional space. The differential electromagnetic equations in fractional space, established in this chapter, provide a basis for application of the concept of fractional space in practical electromagnetic wave propagation and scattering problems in fractal media. In Sect. 2.1 a review of already existing study to construct a generalized Laplacian operator using integration in D-dimensional fractional space is briefly described. In Sect. 2.2, fractional space generalization of the Del operator, written as ∇ D , and its related differential operators ( i.e., gradient, divergence and curl) in vector calculus is obtained. In Sect. 2.3, a novel fractional space generalization of differential Maxwell’s equations is presented. In Sect. 2.4, fractional space generalization of the Laplace and Poisson’s equations is established in addition to fractional space generalization of potentials for static field. In Sect. 2.5, potentials for time varying fields in fractional space are derived. In Sect. 2.6, the Helmholtz’s equation in fractional space is established. Finally, this chapter is summarized in Sect. 2.7.

2.1 Fractional Space Generalization of Laplacian Operator In [1] a formalism is provided for integration on D-dimensional fractional space. According to this formalism, the integration of radially symmetric function f (r ) in a D-dimensional fractional space is given by [1]: ∞ dx0 f (r (x0 , x1 )) = dr W (r ) f (r ) (2.1) 0

where r (x0 , x1 ) is the distance between two points x0 and x1 , and weight W (r ) given by W (r ) = σ (D)r D−1 M. Zubair et al., Electromagnetic Fields and Waves in Fractional Dimensional Space, SpringerBriefs in Applied Sciences and Technology, DOI: 10.1007/978-3-642-25358-4_2, © The Author(s) 2012

(2.2) 7

8

2 Differential Electromagnetic Equations in Fractional Space

with σ (D) =

2π D/2 Γ (D/2)

(2.3)

From this a single variable Laplacian operator is derived in D-dimensional fractional space as: 2 ∂ D−1 ∂ 2 ∇ D f (r ) = f (r ), 0 < D ≤ 1 (2.4) + ∂r 2 r ∂r In Eq. 2.4 and throughout the discussion, the subscript D is used to emphasize the dimension of space in which this operator is defined. An extension of formalism in Eq. 2.1 to two variable integration yields an expression for a two-coordinate Laplacian operator in fractional space. 2 = ∇D

∂2 ∂2 D−2 ∂ + + , 0 1, ignoring terms involving second or higher degree of x in denominator, leads to the following form: |∇ D | =

∂ 1 D−1 + ∂x 2 x

(2.10)

From (2.8) and (2.10), Del operator in single variable x with fractional dimension D is given by: ∂ 1 D−1 ∇D = + xˆ (2.11) ∂x 2 x Extending above procedure to three variable case for | x |, | y |, | z | 1 we get Del operator ∇ D in fractional space as follows: ∂ 1 α1 − 1 1 α2 − 1 ∂ + + xˆ + yˆ ∇D = ∂x 2 x ∂y 2 y ∂ 1 α3 − 1 + + zˆ (2.12) ∂z 2 z where, parameters (0 < α1 ≤ 1, 0 < α2 ≤ 1 and 0 < α3 ≤ 1) are used to describe the measure distribution of space where each one is acting independently on a single coordinate and the total dimension of the system is D = α1 + α2 + α3 . It is important to mention that Eq. 2.12 and all differential operators presented in later sections are valid in far-field region only i.e (|x|, |y|, |z| 1) because of the first order approximation given by (2.10). Clearly, for three dimensional space (D = 3), if we set α1 = α2 = α3 = 1 in (2.12), the fractional Del operator ∇ D reduces to the classical Del operator ∇ [3] in Euclidean space.

10

2 Differential Electromagnetic Equations in Fractional Space

2.2.2 Gradient Operator in Fractional Space The gradient of a scalar field ψ in fractional space is a vector that represents both the magnitude and the direction of maximum space rate of increase of ψ in fractional space. Using (2.12) the modified form of the gradient of scalar field ψ, written as grad D ψ, in far-field region in the fractional space is given as: ∂ψ ∂ψ 1 (α1 − 1)ψ 1 (α2 − 1)ψ grad D ψ = ∇ D ψ = + + xˆ + yˆ ∂x 2 x ∂y 2 y ∂ψ 1 (α3 − 1)ψ + + zˆ (2.13) ∂z 2 z

2.2.3 Divergence Operator in Fractional Space From (2.12) a generalized form of divergence of a vector F = Fx xˆ + Fy yˆ + Fz zˆ at point P(x0 , y0 , z 0 ) in far-field region in the fractional space is written as div D F and is given by div D F = ∇ D · F =

∂ Fy ∂ Fx 1 (α1 − 1)Fx 1 (α2 − 1)Fy + + + ∂x 2 x ∂y 2 y 1 (α3 − 1)Fz ∂ Fz + + ∂z 2 z

(2.14)

2.2.4 Curl Operator in Fractional Space The modified form of curl of a vector F = Fx xˆ + Fy yˆ + Fz zˆ at point P(x 0 , y0 , z 0 ) in far-field region in the fractional space is written as curl D F and using (2.12) it is given by curl D F = ∇ D × F ∂ Fz + = ∂y ∂ Fx + ∂z ∂ Fy + ∂x

∂ Fy 1 (α2 − 1)Fz + − 2 y ∂z ∂ Fz 1 (α3 − 1)Fx + − 2 z ∂x ∂ Fx 1 (α1 − 1)Fy + − 2 x ∂y

1 (α3 − 1)Fy xˆ 2 z 1 (α1 − 1)Fz + yˆ 2 x 1 (α2 − 1)Fx + zˆ 2 y

(2.15)

or curl D F = ∇ D

yˆ xˆ zˆ ∂ 1 α1 − 1 ∂ 1 α2 − 1 ∂ 1 α3 − 1 + + + × F= (2.16) ∂x 2 x ∂y 2 y ∂z 2 z Fx Fy Fz

2.3 Fractional Space Generalization of Differential Maxwell’s Equations

11

2.3 Fractional Space Generalization of Differential Maxwell’s Equations The Maxwell’s equations are the fundamental equations describing the behavior of electric and magnetic fields. In classical electromagnetic theory following quantities are dealt with: E = electric field intensity (V /m) B = magnetic field intensity (A/m) D = electric flux density (C/m 2 ) B = magnetic flux density (W/m 2 ) J = electric current density (A/m 2 ) ρv = electric charge density (C/m 3 ) with B = μH and D = εE, where μ and ε are permeability and permittivity of the medium, respectively. All of these quantities are functions of space variables x, y, z and tim t. The basic classical Maxwell’s equations in differential form in Euclidean space are [4]: ∇ · D = ρv

(2.17)

∇ ·B=0

(2.18)

∇ × E=−

∂B ∂t

∇ × H=J+

(2.19)

∂D ∂t

(2.20)

Also the continuity equation ∇ ·J=−

∂ρv ∂t

(2.21)

is implicit in Maxwell’s equations. Now we wish to have a generalized form of Maxwell’s equations in D-dimensional fractional space. From the results of Sect. 3, we are now able to write differential form of Maxwell’s equations in far-field region in the fractional space as follows: div D D = ρv

(2.22)

div D B = 0

(2.23)

curl D E = −

∂B ∂t

curl D H = J +

∂D ∂t

(2.24) (2.25)

12

2 Differential Electromagnetic Equations in Fractional Space

and the continuity equation in fractional space as: div D J = −

∂ρv ∂t

(2.26)

where, div D and curl D are defined in Eqs. 2.14–2.16. Equations 2.22–2.25 provide generalization of classical Maxwell’s equations form integer dimensional Euclidean space to a non-integer dimensional fractional space. For D = 3, these fractional equations can be reduced to classical Maxwell’s equations in Euclidean space. In phasor form, assuming a time factor e jωt , Maxwell’s equations in fractional space are given by replacing ∂t∂ with jω [4] as below: div D Ds = ρvs

(2.27)

div D Bs = 0

(2.28)

curl D Es = − jωBs

(2.29)

curl D Hs = Js + jωDs

(2.30)

and the phasor form of continuity equation in fractional space as: div D Js = − jωρvs

(2.31)

where, Ds , Bs , Es , Hs , Js , ρvs represent the phasor form of instantaneous quantities D, B, E, H, J and ρv , respectively.

2.4 Fractional Space Generalization of Potentials for Static Fields, Poisson’s and Laplace’s Equations From Maxwell’s equations in previous section, it is shown that the behavior of electrostatic field in fractional space can be described by two differential equations: div D E =

ρv ε0

curl D E = 0

(2.32) (2.33)

where, ε0 is permittivity of free space. Equation 2.33 being equivalent to the statement that E is the gradient of a scalar function, the scalar potential for electric field ψ. Because curl D (−grad D ψ) = 0

(2.34)

2.4

Fractional Space Generalization

13

so, E = −grad D ψ

(2.35)

A detailed proof of Eq. 2.34 is provided in Appendix A. Equations 2.32 and 2.35 can be combined into one partial differential equation for the single function ψ(x, y, z) as follows: div D grad D ψ =

ρv ε0

(2.36)

2 ψ, so finally we get As div D grad D ψ = ∇ D 2 ∇D ψ=

ρv ε0

(2.37)

2 is scalar Laplacian operator in fractional space given by (2.6). Equation where ∇ D 2.37 is called Poisson’s equation in fractional space. In regions of space that lack a charge density, the scalar potential ψ satisfies the Laplace’s equation given by: 2 ∇D ψ =0

(2.38)

Equations 2.37–2.38 are important in solving practical electrostatic problems in fractional space. From Maxwell’s equations in last section, it is shown that the behavior of magnetostatic field in fractional space can be described by two differential equations: div D B = 0

(2.39)

curl D H = J

(2.40)

From Eq. 2.40 we say that in problems concerned with finding the magnetic fields in a current free region, the curl D of magnetic field H is zero. Any vector with zero curl D may be represented as the grad D of a scalar (see e.g., Eq. 2.34). Thus, the magnetic field for points in such regions can be expressed as H = −grad D ψm

(2.41)

where, ψm (in amperes) is the magnetic scalar potential and the minus sign is taken to complete the analogy with electrostatic field in (2.35). From (2.39), the divergence of B is zero everywhere, so using (2.39) and (2.41) div D (μgrad D ψm ) = 0

(2.42)

Thus for a homogenous medium in fractional space the magnetic scalar potential ψm satisfies the Laplace equation:

14

2 Differential Electromagnetic Equations in Fractional Space 2 ∇D ψm = 0

(2.43)

From (2.39) we know that for magnetostatic field div D B = 0. Also we know that div D curl D A = 0

(2.44)

In order to satisfy (2.39) and (2.44) simultaneously, we can define vector magnetic potential A (in webers/meter) such that B = curl D A

(2.45)

Now if we substitute (2.45) into (2.40) we get curl D curl D A = μJ

(2.46)

This may be considered as differential equation relating A to the current density J. Using vector identity 2 A curl D curl D A = grad D (div D A) − ∇ D

(2.47)

div D A = 0

(2.48)

2 ∇D A = −μJ

(2.49)

with

in (2.46) we get

This is a vector equivalent of Poisson’s equation in (2.37). It includes three component scalar equations which are exactly of the poisson form.

2.5 Fractional Space Generalization of Potentials for Time-Varying Fields A we have seen, in Maxwell’s equations fields are related to each other and sources as well. But sometimes it is helpful to introduce some intermediate functions, known as potential functions, which are directly related to sources and from which we can drive fields [4]. Such function are found useful for static fields as well (see e.g., Eqs. 2.35, 2.41, 2.45). From (2.45) we have B = curl D A. This relation may now be substituted into Maxwell’s equation (2.24) to get ∂A =0 (2.50) curl D E + ∂t

2.5

Fractional Space Generalization of Potentials for Static Fields

15

Equation 2.50 states that curl D of a certain quantity is zero. But this condition allows a vector to be derived as a grad D of a scalar ψ. ∂A = −grad D ψ ∂t ∂A E = −grad D ψ − ∂t

E+

(2.51) (2.52)

Equation 2.45 and 2.52 are the valid relationships between fields and potential functions A and ψ. We substitute (2.52) into (2.22), to obtain 2 −∇ D ψ−

ρv ∂(div D A) = ∂t ε

(2.53)

Then by substituting (2.45) and (2.52) into (2.53), we get

∂ψ ∂ 2A curl D curl D A = μJ + με −grad D − 2 ∂t ∂t

(2.54)

Using the vector identity (2.45) and choosing div D A = −με

∂ψ ∂t

(2.55)

Equation 2.53 and 2.54 can be reduced to 2 ψ − με ∇D

∂ 2ψ ρv =− 2 ∂t ε

(2.56)

2 ∇D A − με

∂ 2A = −μJ ∂t 2

(2.57)

Thus the potential functions A and ψ, defined in terms of sources J and ρv by the Eqs. 2.56 and 2.57 in fractional space, may be used to drive electric and magnetic fields using (2.45) and (2.52).

2.6 Fractional Space Generalization of the Helmholtz’s Equation From Eqs. 2.24 and 2.25, using B = μH and D = εE, we finally obtain ∂H ∂t ∂E curl D H = J + ε ∂t curl D E = −μ

Taking curl D of Eq. 2.58 on both sides and using (2.59) gives

(2.58) (2.59)

16

2 Differential Electromagnetic Equations in Fractional Space

curl D curl D E = −μ

∂E ∂ J+ε ∂t ∂t

(2.60)

This result can be simplified using (2.47) and (2.26) in (2.60) as : ∂ 2E 1 ∂J grad D ρv + μ + με 2 ε ∂t ∂t For source-free region (ρv = 0, J = 0) (2.61) becomes 2 ∇D E=

(2.61)

∂ 2E =0 (2.62) ∂t 2 Equation 2.62 is the Helmholtz’s equation, or wave equation, for E in fractional space. An identical equation for H in fractional space can also be derived in the same manner: 2 ∇D E − με

2 H − με ∇D

∂ 2H =0 ∂t 2

(2.63)

2.7 Summary A novel fractional space generalization of the differential electromagnetic equations, that is helpful in studying the behavior of electric and magnetic fields in fractal media, is provided. A new form of vector differential operator Del, written as ∇ D , and its related differential operators is formulated in fractional space. Using these modified vector differential operators, the classical Maxwell’s electromagnetic equations have been worked out. The Laplace’s, Poisson’s and Helmholtz’s equations in fractional space are derived by using modified vector differential operators. Also a new fractional space generalization of potentials for static and time-varying fields is presented. For all investigated cases, when integer dimensional space is considered, the classical results can be recovered. The provided fractional space generalization of differential electromagnetic equations is valid in far-field region only. The differential electromagnetic equations in fractional space, established in this work, provide a basis for application of the concept of fractional space in practical electromagnetic wave propagation and scattering phenomenon in far-field region in any fractal media.

References 1. F.H. Stillinger, Axiomatic basis for spaces with noninteger dimension. J. Math. Phys. 18(6), 1224–1234 (1977) 2. C. Palmer, P.N. Stavrinou, Equations of motion in a noninteger-dimension space. J Phys A 37, 6987–7003 (2004) 3. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables. (Department of Commerce, U.S., 1972) 4. C.A. Balanis, Advanced Engineering Electromagnetics. (Wiley, New York, 1989)

Chapter 3

Potentials for Static and Time-Varying Fields in Fractional Space

In this chapter, a fractional space generalization of potentials for static and timevarying fields is discussed. The fractional space generalization of static and time-varying potentials, provided in this chapter, can be used to study electrostatic problems in fractal media. In Sect. 3.1, electrostatic potential in D-dimensional fractional space is studied. In Sect. 3.2, time-varying auxiliary potential is studied in fractional space.

3.1 Electrostatic Potential in Fractional Space In this section, an exact solution of the Laplace’s equation for electrostatic potential in D-dimensional fractional space is presented. As an application, the electrostatic potential inside the rectangular box with surfaces held at constant potentials is obtained in fractional pace. It is also shown that for integer value of dimension D, the classical results are recovered. The obtained solution can be used to study complex electrostatic problems in fractal media. In Sect. 3.1.1, we investigate full analytical solution to the Laplace’s equation in D-dimensional fractional space, where the parameter D is used to describe the measure distribution of space. In Sect. 3.1.2, the electrostatic potential inside the rectangular box with surfaces held at constant potentials in fractional space is obtained. Finally, in Sect. 3.1.3, results are summarized.

3.1.1 An Exact Solution of the Laplace’s Equation in D-dimensional Fractional Space The Laplace’s equation in D-dimensional fractional space can describe complex physical phenomenon. Laplace’s equation in fractional space is given by: 2 ∇D Ψ =0

(3.1.1)

M. Zubair et al., Electromagnetic Fields and Waves in Fractional Dimensional Space, SpringerBriefs in Applied Sciences and Technology, DOI: 10.1007/978-3-642-25358-4_3, © The Author(s) 2012

17

18

3 Potentials for Static and Time-Varying Fields

2 is the scalar laplacian operator in where ψ is the electrostatic potential and ∇D D-dimensional fractional space given as follows [1]. 2 = ∇D

∂2 α1 − 1 ∂ α2 − 1 ∂ α3 − 1 ∂ ∂2 ∂2 + + + + + ∂x2 x ∂x ∂y2 y ∂y ∂z 2 z ∂z

(3.1.2)

where, three parameters (0 < α1 ≤ 1, 0 < α2 ≤ 1 and 0 < α3 ≤ 1) are used to describe the measure distribution of space where each one is acting independently on a single coordinate and the total dimension of the system is D = α1 + α2 + α3 . It is obvious that for three dimensional space (D = 3), if we set α1 = α2 = α3 = 1 2 reduces to the classical Laplacian in (3.1.2), the fractional Laplacian operator ∇D operator ∇ 2 in Euclidean space. In this section we present an exact solution of the Laplace equation in (3.1.2) for electrostatic potential in D-dimensional fractional space. In expanded form, (3.1.1) can be written as:

∂2 α1 − 1 ∂ α2 − 1 ∂ α3 − 1 ∂ ∂2 ∂2 + + + + + 2 2 2 ∂x x ∂x ∂y y ∂y ∂z z ∂z

ψ =0

(3.1.3)

Equation (3.1.6) is separable using separation of variables. We consider ψ(x, y, z) = f (x)g(y)h(z)

(3.1.4)

the resulting ordinary differential equations are obtained as follows: α1 − 1 d d2 2 + + α f (x) = 0 dx2 x dx 2 d α2 − 1 d 2 + β g(y) = 0 + dy2 y dy 2 α3 − 1 d d 2 + − γ h(z) = 0 d z2 z dz

(3.1.5b)

α2 + β 2 = γ 2

(3.1.6)

(3.1.5a)

(3.1.5c)

where, in addition,

Equation (3.1.5a) through (3.1.5c) are all of the same form; solution for any one of them can be replicated for others by inspection.We choose to work first with f (x). We write (3.1.5a) as d d2 + α2 x f = 0 x 2 +a dx dx

(3.1.7)

where, a = α1 − 1. Equation (3.1.7) is reducible to Bessel’s equation under substitution f = xv1 ξ as follows:

3.1 Electrostatic Potential in Fractional Space

d2 d |1 − a| 2 2 2 x +x + (α x − v1 ) ξ = 0, v1 = dx2 dx 2 2

19

(3.1.8)

The solution of Bessel’s equation in (3.1.8) is given as [2] ξ = C1 Jv1 (αx) + C2 Yv1 (αx)

(3.1.9)

where, Jv1 (αx) is referred to as Bessel function of the first kind of order v1 , Yv1 (αx) as the Bessel function of the second kind of order v1 . Finally the solution of (3.1.5a) becomes α1 f (x) = xv1 C1 Jv1 (αx) + C2 Yv1 (αx) , v1 = 1 − (3.1.10) 2 Similarly, the solutions to (3.1.5b) and (3.1.5c) are obtained as α2 (3.1.11) g(y) = yv2 C3 Jv2 (βy) + C4 Yv2 (βy) , v2 = 1 − 2 h(z) = z v3 C5 Jv3 (−jγ z) + C6 Yv3 (−jγ z) α3 = z v3 C5 Jv3 (−j α 2 + β 2 z) + C6 Yv3 (−j α 2 + β 2 z) , v3 = 1 − 2 (3.1.12) From (3.1.4) and (3.1.10) through (3.1.13), the solution of (3.1.3) have the form ψ(x, y, z) = xv1 yv2 z v3 C1 Jv1 (αx) + C2 Yv1 (αx) × C3 Jv2 (βy) + C4 Yv2 (βy) × C5 Jv3 (−j α 2 + β 2 z) + C6 Yv3 (−j α 2 + β 2 z) (3.1.13) where, C1 through C6 are constant coefficients. Also α and β can be determined by imposing boundary conditions on potential. This solution can be used to study the electrostatic field in a non-integer dimensional fractal media.

3.1.2 Electrostatic Potential Inside a Rectangular Box in Fractional Space In the present section we determine the potential inside a rectangular box in D-dimensional fractional space with dimensions (a, b, c) in the (x, y, z) directions. All surfaces of the box are kept at zero potential, except the surface z = c, which is at potential V (x, y). Starting from the requirement that ψ = 0 at x = 0, y = 0, z = 0, it is easy to see that the required forms of f (x), g(y), h(z) are f (x) = xv1 Jv1 (αx)

(3.1.14a)

g(y) = yv2 Jv2 (βy)

(3.1.14b)

20

3 Potentials for Static and Time-Varying Fields

h(z) = z v3 Jv3 (−j α 2 + β 2 z)

(3.1.14c)

To have ψ = 0 at x = a and y = b, we must have αa =

π(4n + 2v1 + 3) 4

(3.1.15a)

βb =

π(4m + 2v2 + 3) 4

(3.1.15b)

With the definitions, αn =

π(4n + 2v1 + 3) 4a

π(4m + 2v2 + 3) 4b 4m + 2v2 + 3 2 π 4n + 2v1 + 3 2 = + 4 a b

(3.1.16a)

βm =

(3.1.16b)

γmn

(3.1.16c)

The partial potential ψmn satisfying all boundary conditions except one, can be written as: ψmn = xv1 yv2 z v3 Jv1 (αn x)Jv2 (βm y)Jv3 (−jγmn z)

(3.1.17)

And the potential can expanded in terms of these ψmn with initial arbitrary coefficients (to be chosen to satisfy final boundary condition): ψ(x, y, z) =

∞

Amn xv1 yv2 z v3 Jv1 (αn x)Jv2 (βm y)Jv3 (−jγmn z)

(3.1.18)

n,m=1

And the final boundary condition ψ = V (x, y) at z = c: V (x, y) =

∞

Amn xv1 yv2 cv3 Jv1 (αn x)Jv2 (βm y)Jv3 (−jγmn c)

(3.1.19)

n,m=1

so, the constant coefficients Amn are given by: a b 4 Amn = dx dyV (x, y)xv1 yv2 Jv1 (αn x)Jv2 (βm y) (3.1.20) abcv3 Jv3 (−jγmn c) 0 0 Equation (3.1.18) provides the required solution in fractional space. Now, if we take D = 3 i.e., α1 = α2 = α3 = 1 in (3.1.18) and use Bessel functions of fractional order then the classical results given by Jackson [3], for the same problem in Euclidean space, can be recovered.

3.1 Electrostatic Potential in Fractional Space

21

3.1.3 Summary An exact solution of the Laplace’s equation for electrostatic potential in D-dimensional fractional space is obtained. The electrostatic potential inside the rectangular box with surfaces held at constant potentials is obtained in fractional space. It is also shown that for integer value of dimension D, the classical results can be recovered. The obtained solution can be used to study complex electrostatic problems in fractal media.

3.2 Time-Varying Potentials in Fractional Space The procedure for analysis of radiation problems is to specify sources and get the fields radiated by the sources. For analysis of radiation problems in fractional space we have to introduce auxiliary potential function A (magnetic vector potential) and F (electric vector potential). The fractional space generalization of the relation between auxiliary potential functions and sources is given by inhomogeneous vector potential wave equations. In this Section, a novel exact solution of the inhomogeneous vector potential wave equations in D-dimensional fractional space is presented. It is also shown that for integer values of dimension D, the classical results are recovered. The solution of inhomogeneous vector potential wave equation in fractional space is useful to study the radiation phenomenon in fractal media. In Sect 3.2.1, we investigate full analytical solution of the inhomogeneous vector potential wave equation in D-dimensional fractional space, where the parameter D is used to describe the measure distribution of space, also the solution of inhomogeneous vector potential wave equation in integer-dimensional space is justified from the results obtained. Finally, results are summarized drawn in Sect. 3.2.2.

3.2.1 Inhomogeneous Vector Potential Wave Equation in D-dimensional Fractional Space The procedure for analysis of radiation problems is to specify sources and get the fields radiated by the sources. For analysis of radiation problems in fractional space we have to introduce auxiliary potential function A (magnetic vector potential) and F (electric vector potential). The fractional space generalization of the relation between auxiliary potential functions and sources is given by inhomogeneous vector potential wave equations as below: 2 A + k 2 A = −μJ ∇D

(3.2.1)

2 ∇D F + k 2 F = −εM

(3.2.2)

22

3 Potentials for Static and Time-Varying Fields

2 is the modified laplacian operator in D-dimensional where, k 2 = ω2 με and ∇D fractional space defined in Equation (3.1.2) and J, M are harmonic electric and magnetic currents. In spherical coordinates fractional Laplacian operator becomes: 2 ∇D

2 ∂ ∂2 D−1 ∂ D−2 ∂ 1 = 2+ + + ∂r r ∂r r2 ∂θ 2 tan θ ∂θ 2 1 D−3 ∂ ∂ + 2 + 2 r sin θ ∂φ tan φ ∂φ

(3.2.3)

where, 2 < D ≤ 3. In this section we present a novel exact solution of the inhomogeneous vector potential wave equations in (3.2.1) and (3.2.2) in D-dimensional fractional space. Once the solution to any one of equation (3.2.1) and (3.2.2) in fractional space is known, the solution to the other can be written by duality principle. To drive the solution to (3.2.1), we assume a source with current density Jz , which in limit is an infinitesimal point source, is placed at origin of a x, y, z coordinate system. Since the current density is directed along z-axis, only an Az component will exist. Thus, using (3.2.1) 2 Az + k 2 Az = −μJz ∇D

(3.2.4)

At point removed from the source (Jz = 0), the wave equation reduces to 2 Az + k 2 Az = 0 ∇D

(3.2.5)

Since in the limit the source is a point, it requires Az as function of r in spherical coordinates (i.e., Az is not a function of θ and φ. Thus, using the definition of Laplacian operator from (3.2.3) we get (3.2.5) in expanded form as:

∂2 D−1 ∂ 2 + + k Az (r) = 0 ∂r2 r ∂r

(3.2.6)

where, 2 < D ≤ 3. The partial derivatives are replaced with ordinary derivative because Az is a function of r only. Equation (3.2.6) is reducible to Bessel’s equation under substitution Az (r) = xn ξ as follows: r2

d2 d D 2 2 2 + r r − n ) ξ = 0, n = 1 − + (k 2 dr dr 2

(3.2.7)

The solution of Bessel’s equation in (3.2.7) is given as [2] ξ = C1 Hn(1) (kr) + C2 Hn(2) (kr) (1)

(3.2.8) (2)

where, Hn (kr) is referred to as Hankel function of the first kind of order n, Hn (kr) as the Hankel function of the second kind of order n. Finally, the solution of (3.2.6) becomes

3.2 Time-Varying Potentials in Fractional Space

23

D Az = rn C1 Hn(1) (kr) + C2 Hn(2) (kr) , n = 1 − 2

(3.2.9)

So the two independent solutions of (3.2.6) are Az1 = C1 rn Hn(1) (kr)

(3.2.10a)

Az2 = C2 rn Hn(2) (kr)

(3.2.10b)

Equations (3.2.10a, b) represent inward and outward going waves (assuming a time dependency ejωt ). For this problem source is located at origin and the waves are going outward. So, we choose the solution of (3.2.6) as Az = C1 rn Hn(2) (kr)

(3.2.11)

In the static case (ω = 0, k = 0), (3.2.11) simplifies to Az = C1 rn Hn(2) (0)

(3.2.12)

Thus, at points removed from the source, the time-varying and static solutions of (3.2.11) and (3.2.12) differ only by the argument of Hankel function from kr to zero; or the time-varying solution can be found by multiplying static form with √ (2) rHn (kr). In the presence of the source (Jz = 0) an (k = 0) the wave equation (2.2.4) is reduced to 2 ∇D Az = −μJz

(3.2.13)

This equation is known as Poisson’s equation in fractional space. The well known form of Poisson’s equation relating the scalar field ψ to the electric charge density ρ is given by 2 ψ= ∇D

ρ ε

and the solution of (3.2.14) is given by [4] as follow: ρ 23−D Γ (3/2) ψ= dV Γ (D/2) rD−2

(3.2.14)

(3.2.15)

where, r is the distance from any point on charge density to observation point. As (3.2.13) is similar to (3.2.14), so its solution is same as in (3.2.15). μ 23−D Γ (3/2) Jz dV (3.2.16) Az = D−2 4π Γ (α/2) r using analogy between (3.2.11) and (3.2.12), the time-varying solution of (3.2.4) is given by

24

3 Potentials for Static and Time-Varying Fields

Az =

μ 23−D Γ (3/2) 4π Γ (D/2)

(2)

Jz

Hn (kr) dV rD−5/2

(3.2.17)

If the current were in the x- and y-directions (Jx ,Jy ), the wave equations for each would reduce to have same form as (3.2.4) and will possess the same solutions as in equation (3.2.17). Finally, we have the solution of vector wave equation in (3.2.4) as: Hn(2) (kr) μ 23−D Γ (3/2) J D−5/2 d V A= (3.2.18) 4π Γ (D/2) r If the source is removed from origin and placed at position represented by the primed coordinates (x , y , z ), (3.2.18) can be written as: (2) μ 23−D Γ (3/2) Hn (kR) A= (3.2.19) J(x , y , z ) D−5/2 d V 4π Γ (D/2) R where, R is the distance between any point in the source to observation point, Γ (x) (2) is the gamma function and Hn (kR) denotes the Hankel function of second kind of order n representing outward waves from source point. Now, for validation of our provided solution in (3.2.19), we get vector potential A from our solution by substituting D = 3 in (3.2.19). For D = 3, we have n = −1/2. Using following Hankel function of fractional order [5] 2 −jx 2 (3.2.20) e H 1 (x) = πx 2 Equation (3.2.19) gets reduced to e−jkR μ A = C dV J(x , y , z ) 4π R

(3.2.21)

where, C is a constant term. Equation (3.2.21) is in exact agreement with the solution provided in [6] for Euclidean space. In a similar fashion, we can show the solution of (3.2.2) as (2) ε 23−D Γ (3/2) Hn (kR) F= (3.2.22) M(x , y , z ) D−5/2 d V 4π Γ (D/2) R For D = 3, using Hankel function of fractional order given by (3.2.20), finally equation (3.2.22) gets reduced to e−jkR ε (3.2.23) dV M(x , y , z ) F=C 4π R Equation (3.2.23) is in exact agreement with the solution provided in [6] for Euclidean space. The fractional space solutions of inhomogeneous vector potential wave equation, given by (3.2.19) and (3.2.22), can be used to solve complex radiation problems in fractional space.

3.2 Time-Varying Potentials in Fractional Space

25

3.2.2 Summary An exact solution of the inhomogeneous vector potential wave equation in D-dimensional fractional space is presented. It is also shown that for integer values of dimension D, the classical results are recovered. The solution of inhomogeneous vector potential wave equation in fractional space is useful to study the radiation phenomenon in fractal media.

References 1. C. Palmer, P.N. Stavrinou, Equations of motion in a noninteger-dimension space. J. Phys. A 37, 6987–7003 (2004) 2. A.D. Polyanin, V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. (CRC Press, Boca Raton, 2003) 3. J.D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley, New York, 1999) 4. S. Muslih, D. Baleanu, Fractional Multipoles in fractional space. Nonlinear Anal. Real World Appl. 8, 198–203 (2007) 5. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables (U.S. Department of Commerce, USA, 1972) 6. C.A. Balanis, Antenna Theory: Analysis and Design (Wiley, New York, 1982)

Chapter 4

Electromagnetic Wave Propagation in Fractional Space

The wave equation has very important role in many areas of physics. It has a fundamental meaning in classical as well as quantum field theory. With this view, one is strongly motivated to discuss solutions of the wave equation in all possible situations. The wave equation in fractional space can effectively describe the wave propagation phenomenon in fractal media. In this chapter, exact solutions of different forms of wave equation in D-dimensional fractional space are provided, which describe the phenomenon of electromagnetic wave propagation in fractional space. In Sect. 4.1, the general plane wave solutions in fractional space are provided for lossless medium case. In Sect. 4.2, the general plane wave solutions in fractional space are provided for lossy medium case. In Sect. 4.3, the general cylindrical wave solutions in fractional space are provided by solving cylindrical wave equation in D-dimensional fractional space. In Sect. 4.4, the general spherical wave solutions in fractional space are provided by solving spherical wave equation in D-dimensional fractional space.

4.1 General Plane Wave Solutions in Fractional Space: Lossless Medium Case In Sect. 4.1.1, we investigate full analytical solution of wave equation in D-dimensional fractional space, where three parameters are used to describe the measure distribution of space. In Sect. 4.1.2, solution of wave equation in integerdimensional space is obtained from the results of previous section. Finally, in Sect. 4.1.3, major results are summarized.

4.1.1 General Plane Wave Solutions in Fractional Space For source-free and lossless media, the vector wave equations for the complex electric and magnetic field intensities are given by the Helmholtz’s equation as follows [1]. M. Zubair et al., Electromagnetic Fields and Waves in Fractional Dimensional Space, SpringerBriefs in Applied Sciences and Technology, DOI: 10.1007/978-3-642-25358-4_4, © The Author(s) 2012

27

28

4 Electromagnetic Wave Propagation in Fractional Space 2 ∇D E + β2E = 0

(4.1.1a)

+β H =0

(4.1.1b)

2 H ∇D

2

where, β 2 = ω2 με. Time dependency e jwt has been suppressed throughout the 2 is the scalar Laplacian operator in D dimensional fractional discussion. Here ∇ D space and is defined as follows [2]. 2 ∇D =

∂2 ∂2 ∂2 α1 − 1 ∂ α2 − 1 ∂ α3 − 1 ∂ + + + + + 2 2 2 ∂x x ∂x ∂y y ∂y ∂z z ∂z

(4.1.2)

Eq. (4.1.2) uses three parameters (0 < α1 ≤ 1, 0 < α2 ≤ 1 and 0 < α3 ≤ 1) to describe the measure distribution of space where each one is acting independently on a single coordinate and the total dimension of the system is D = α1 + α2 + α3 . Once the solution to any one of Eqs. (4.1.1a and 4.1.1b) in fractional space is known, the solution to the other can be written by an interchange of E with H or H with E due to duality. We will examine the solution for E. In rectangular coordinates, a general solution for E can be written as E(x, y, z) = aˆ x E x (x, y, z) + aˆ y E y (x, y, z) + aˆ z E z (x, y, z)

(4.1.3)

Substituting (4.1.3) into (4.1.1a) we can write that 2 ∇D (aˆ x E x + aˆ y E y + aˆ z E z ) + β 2 (aˆ x E x + aˆ y E y + aˆ z E z ) = 0

(4.1.4)

which reduces to three scalar wave equations as follows: 2 E x (x, y, z) + β 2 E x (x, y, z) = 0 ∇D

(4.1.5a)

2 E y (x, ∇D 2 ∇D E z (x,

y, z) + β E y (x, y, z) = 0

(4.1.5b)

y, z) + β 2 E z (x, y, z) = 0

(4.1.5c)

2

Eq. (4.1.5a) through (4.1.5c) are all of the same form; solution for any one of them in fractional space can be replicated for others by inspection.We choose to work first with E x as given by (4.1.5a). In expanded form (4.1.5a) can be written as α1 − 1 ∂ E x α2 − 1 ∂ E x ∂ 2 Ex ∂ 2 Ex ∂2 Ex + + + + 2 2 ∂x x ∂x ∂y y ∂y ∂z 2 α3 − 1 ∂ E x + + β2 Ex = 0 z ∂z

(4.1.6)

Equation (4.1.6) is separable using separation of variables. We consider E x (x, y, z) = f (x)g(y)h(z) the resulting ordinary differential equations are obtained as follows:

(4.1.7)

4.1 General Plane Wave Solutions in Fractional Space: Lossless Medium Case

29

d2 α1 − 1 d 2 + + βx f = 0 dx2 x dx 2 d α2 − 1 d 2 + + β y g =0 dy 2 y dy 2 d α3 − 1 d 2 + β + z h =0 dz 2 z dz

(4.1.8b)

βx2 + β y2 + βz2 = β 2

(4.1.9)

(4.1.8a)

(4.1.8c)

where, in addition,

Equation (4.1.9) is referred to as constraint equation. In addition βx , β y , βz are known as wave constants in the x, y, z directions, respectively, which will be determined using boundary conditions. Equation (4.1.8a) through (4.1.8c) are all of the same form; solution for any one of them can be replicated for others by inspection.We choose to work first with f (x). We write (4.1.8a) as x

d d2 +a + βx2 x 2 dx dx

f =0

(4.1.10)

where, a = α1 − 1. Equation (4.1.10) is reducible to Bessel’s equation under substitution f = x n ξ as follows:

d |1 − a| d2 2 2 2 +x + (βx x − n ) ξ = 0, n = x 2 dx dx 2 2

(4.1.11)

The solution of Bessel’s equation in (4.1.11) is given as [3] ξ = C1 Jn (βx x) + C2 Yn (βx x)

(4.1.12)

where, Jn (βx x) is referred to as Bessel function of the first kind of order n, Yn (βx x) as the Bessel function of the second kind of order n. Finally the solution of (4.1.8a) becomes α1 f (x) = x n 1 C1 Jn 1 (βx x) + C2 Yn 1 (βx x) , n 1 = 1 − 2

(4.1.13)

Similarly, the solutions to (4.1.8b) and (4.1.8c) are obtained as α2 g(y) = y n 2 C3 Jn 2 (β y y) + C4 Yn 2 (β y y) , n 2 = 1 − 2 α3 n3 h(z) = z C5 Jn 3 (βz z) + C6 Yn 3 (βz z) , n 3 = 1 − 2

(4.1.14) (4.1.15)

From (4.1.7) and (4.1.13) through (4.1.15), the solution of (4.1.5a) have the form

30

4 Electromagnetic Wave Propagation in Fractional Space

E x (x, y, z) = x n 1 y n 2 z n 3 C1 Jn 1 (βx x) + C2 Yn 1 (βx x) × C 3 Jn 2 (β y y) + C 4 Yn 2 (β y y) × C5 Jn 3 (βz z) + C 6 Yn 3 (βz z)

(4.1.16)

where, C1 through C 6 are constant coefficients. Similarly, the solutions to (4.1.5b) and (4.1.5c) are obtained as E y (x, y, z) = x n 1 y n 2 z n 3 D1 Jn 1 (βx x) + D2 Yn 1 (βx x) × D3 Jn 2 (β y y) + D4 Yn 2 (β y y) (4.1.17) × D5 Jn 3 (βz z) + D6 Yn 3 (βz z) and

E z (x, y, z) = x n 1 y n 2 z n 3 F1 Jn 1 (βx x) + F2 Yn 1 (βx x) × F3 Jn 2 (β y y) + F4 Yn 2 (β y y) × F5 Jn 3 (βz z) + F6 Yn 3 (βz z)

(4.1.18)

where, D1 through D6 and F1 through F6 are constant coefficients. For e jwt time variations, the instantaneous form E (x, y, z; t) of the vector complex function E(x, y, z) in (4.1.8a) takes the form E (x, y, z; t) = e[{aˆ x E x (x, y, z) + aˆ y E y (x, y, z) + aˆ z E z (x, y, z)}e jwt ] (4.1.19) where E x (x, y, z), E y (x, y, z) and E z (x, y, z) are given by (4.1.16) through (4.1.18). Equation (4.1.19) provides a general plane wave solution in fractional space. This solution can be used to study the phenomenon of electromagnetic wave propagation in any non-integer dimensional space.

4.1.2 Discussion on Fractional Space Solution Equation (4.1.19) is the generalization of the concept of wave propagation in integer dimensional space to the wave propagation in non-integer dimensional space. As a special case, for three-dimensional space, this problem reduces to classical wave propagation concept; i.e., if we set α1 = 1 in Eq. (4.1.13) then n 1 = 12 and it gives 1 (4.1.20) f (x) = x 2 C1 J 1 (βx x) + C2 Y 1 (βx x) 2

2

Using Bessel functions of fractional order [4]: 2 J 1 (x) = sin (x) 2 πx 2 cos (x) Y 1 (x) = − 2 πx

(4.1.21a) (4.1.21b)

4.1 General Plane Wave Solutions in Fractional Space: Lossless Medium Case

31

Eq. (4.1.13) can be reduced to f (x) = C1 sin(βx x) + C2 cos(βx x)

(4.1.22)

where, Ci = Ci πβ2 x , i = 1, 2 Similarly, we set α2 = 1 and α3 = 1 in (4.1.14) and (4.1.15) respectively and using Bessel functions of fractional order in (4.1.21a) through (b), we get g(y) = C3 sin(β y y) + C 4 cos(β y y) h(z) = C 5 sin(βz z) + C6 cos(βz z)

(4.1.23) (4.1.24)

From (4.1.22) through (4.1.24), we get E x (x, y, z) in three-dimensional space (D = 3) as follows E x (x, y, z) = C1 sin(βx x) + C2 cos(βx x) × C 3 sin(β y y) + C 4 cos(β y y) × C 5 sin(βz z) + C6 cos(βz z) (4.1.25) which is comparable to the solution of wave equation in integer dimensional space obtained by Balanis [1]. Similarly, field components E y (x, y, z) and E z (x, y, z) can also be reduced for three-dimensional case. As another special case, if we choose a single parameter for non-integer dimension D where 2 < D ≤ 3, i.e., we take α1 = α2 = 1 so D = α3 + 2. In this case from Eq. (4.1.6) we obtain ∂ 2 Ex ∂2 Ex D − 3 ∂ Ex ∂ 2 Ex + + + + β2 Ex = 0 ∂x2 ∂ y2 ∂z 2 z ∂z

(4.1.26)

Solving this equation by separation of variables leads to the following result E x (x, y, z) = z n [G 1 cos(βx x) + G 2 sin(βx x)] × G 3 cos(β y y) + G 4 sin(β y y) × G 5 Jn (βz z) + G 6 Yn (βz z) where, n = 2 −

D 2.

(4.1.27)

Here if we set D = 3, and using (4.1.21a) and (b), we get

E x (x, y, z) =

2 [G 1 cos(βx x) + G 2 sin(βx x)] πβz × G 3 cos(β y y) + G 4 sin(β y y) × G 5 sin(βz z) + G 6 cos(βz z)

(4.1.28)

where, G 1 through G 6 are constant coefficients. The result obtained in (4.1.28) is comparable to that obtained by Balanis [1] for 3-dimensional space.

32

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.1 Usual wave propagation (D = 3). [This figure was originally published in [5] , reproduced courtesy of The Electromagnetics Academy]

As an example, an infinite sheet of surface current can be considered as a source of plane waves in D-dimensional fractional space. We assume that an infinite sheet of electric surface current density Js = Js0 xˆ exists on the z = 0 plane in free space. Since the sources do not vary with x or y, the fields will not vary with x or y but will propagate away from the source in ±z direction. The boundary conditions to be satisfied at z = 0 are zˆ × (E2 − E1 ) = 0 and zˆ × (H2 − H1 ) = Js0 x, ˆ where E1 , H1 are the fields for z < 0, and E2 , H2 are the fields for z > 0. To satisfy the later boundary condition, H must have a yˆ component. Then for E to be normal to H and zˆ , E must have an xˆ component. Thus, the corresponding wave equation for E and H fields in D-dimensional fractional space where 2 < D ≤ 3 can be written by modifying (4.1.26) as d2 Ex D − 3 d Ex + + β2 Ex = 0 dz 2 z dz d 2 Hy D − 3 d Hy + + β 2 Hy = 0 2 dz z dz

(4.1.29a) (4.1.29b)

Solution of (4.1.29a) and (4.1.29b) takes the similar form as (4.1.27) and under above mentioned boundary conditions the fields will have the following form: Js0 Js Jn (βz z), H1 = yˆ z n 0 Jn (βz z); z < 0 2 2η0 n Js0 n Js0 Yn (βz z); z > 0 Yn (βz z), H2 = − yˆ z E2 = −xˆ z 2 2η0 E1 = −xˆ z n

(4.1.30a) (4.1.30b)

4.1 General Plane Wave Solutions in Fractional Space: Lossless Medium Case

33

Fig. 4.2 Wave propagation in fractional space (D = 2.5). [This figure was originally published in [5] , reproduced courtesy of The Electromagnetics Academy]

where, η0 is wave impedance in free space. Assuming a time dependency e jwt and Js0 = −2 A/m, the solution for the usual wave for z > 0 with D = 3 is shown in Fig. 4.1, which is comparable to well known plane wave solutions in 3-dimensional space [1]. Similarly, for D = 2.5 we have fractal medium wave for z > 0 as shown in Fig. 4.2, where amplitude variations are described in terms of Bessel functions.

4.1.3 Summary General plane wave solution in source-free and lossless media in fractional space is presented by solving vector wave equation in D-dimensional fractional space. When the wave propagates in fractional space, the amplitude variations are described by Bessel functions. The obtained general plane wave solution is a generalization of integer-dimensional solution to a non-integer dimensional space. For all investigated cases when D is an integer-dimension, the classical results are recovered.

4.2 General Plane Wave Solutions in Fractional Space: Lossy Medium Case In this section, an extension of previous work to the case of plane wave propagation in lossy medium in fractional space is presented. The generalized analytical solution investigated in this section have potential applications in electromagnetic wave propagation problems in lossy media present in fractional space.

34

4 Electromagnetic Wave Propagation in Fractional Space

In Sect. 4.2.1 we investigate full analytical solution of Helmholtz’s equation in source-free, lossy media present in D-dimensional fractional space, where three parameters are used to describe the measure distribution of space. In Sect. 4.2.2, solution of wave equation in integer-dimensional space is obtained from the results of previous section. In Sect. 4.2.3, an example of wave propagation in lossy medium due to current sheet as source of plane waves in fractional space is presented. Finally, results are summarized in Sect. 4.2.4.

4.2.1 General Plane Wave Solutions in Lossy Medium in Fractional Space For source-free and lossy media, the vector wave equations for the complex electric and magnetic field intensities are given by the Helmholtz’s equation as follows [1]. 2 E − γ 2E = 0 ∇D 2 ∇D H − γ 2H = 0

(4.2.1) (4.2.2)

γ 2 = jωμ(σ + jωε)

(4.2.3)

γ = α + jβ σ 2 με

α=ω 1+ −1 2 ωε σ 2 με

1+ +1 β=ω 2 ωε

(4.2.4)

where

(4.2.5)

(4.2.6)

In equation (4.2.3) and (4.2.4) γ = propagation constant α = attenuation constant (Np/m) β = phase constant (rad/m) ε = permittivity of medium (H/m) μ = permeability of medium (F/m) σ = conductivity of medium (S/m) Time dependency e jwt has been suppressed throughout the discussion. 2 is the Laplacian operator in D-dimensional fractional In Eqs. (4.2.1) and (4.2.2), ∇ D space and is defined as follows [2]. 2 = ∇D

∂2 ∂2 ∂2 α1 − 1 ∂ α2 − 1 ∂ α3 − 1 ∂ + + + + + ∂x2 x ∂x ∂ y2 y ∂y ∂z 2 z ∂z

(4.2.7)

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

35

Equation (4.2.7) uses three parameters (0 < α1 ≤ 1, 0 < α2 ≤ 1 and 0 < α3 ≤ 1) to describe the measure distribution of space where each one is acting independently on a single coordinate and the total dimension of the system is D = α1 + α2 + α3 . Once the solution to any one of Eq. (4.2.1) and (4.2.2) in fractional space is known, the solution to the other can be written by an interchange of E with H or H with E due to duality. We will examine the solution for E. In rectangular coordinates, a general solution for E can be written as E(x, y, z) = aˆ x E x (x, y, z) + aˆ y E y (x, y, z) + aˆ z E z (x, y, z)

(4.2.8)

Substituting (4.2.8) into (4.2.1) we can write that 2 (aˆ x E x + aˆ y E y + aˆ z E z ) − γ 2 (aˆ x E x + aˆ y E y + aˆ z E z ) = 0 ∇D

(4.2.9)

which reduces to following three scalar wave equations 2 ∇D E x (x, y, z) − γ 2 E x (x, y, z) = 0

(4.2.10)

2 E y (x, y, z) − γ 2 E y (x, y, z) = 0 ∇D

(4.2.11)

2 E z (x, y, z) − γ 2 E z (x, y, z) = 0 ∇D

(4.2.12)

Equation (4.2.10) through (4.2.12) are all of the same form; solution for any one of them in fractional space can be replicated for others by inspection.We choose to work first with E x as given by (4.2.10). In expanded form (4.2.10) can be written as ∂ 2 Ex ∂2 Ex α1 − 1 ∂ E x α2 − 1 ∂ E x + + + ∂x2 x ∂x ∂ y2 y ∂y 2 ∂ Ex α3 − 1 ∂ E x + − γ 2 Ex = 0 + 2 ∂z z ∂z

(4.2.13)

Equation (4.2.13) is separable using separation of variables. We consider E x (x, y, z) = f (x)g(y)h(z)

(4.2.14)

the resulting ordinary differential equations are obtained as follows:

α1 − 1 d d2 2 + − γ f =0 x dx2 x dx 2 α1 − 1 d d + − γ y2 g = 0 2 dy y dy 2 d α1 − 1 d 2 − γ + z h =0 dz 2 z dz

where, in addition,

(4.2.15) (4.2.16) (4.2.17)

36

4 Electromagnetic Wave Propagation in Fractional Space

γx2 + γ y2 + γz2 = γ 2

(4.2.18)

Equation (4.2.18) is referred to as constraint equation. In addition γx , γ y , γz are known as wave constants in the x, y, z directions, respectively, that will be determined using boundary conditions. Equation (4.2.15) through (4.2.17) are all of the same form; solution for any one of them can be replicated for others by inspection.We choose to work first with f (x). We write (4.2.15) as

d d2 − γx2 x x 2 +a dx dx

f =0

(4.2.19)

where, a = α1 − 1. Equation (4.2.19) is reducible to Bessel’s equation under substitution f = x n ζ as follows:

d |1 − a| d2 2 2 2 +x + (−γx x − n ) ζ = 0, n = x dx2 dx 2 2

(4.2.20)

The solution of Bessel’s equation in (4.2.20) is given as [3] ζ = C 1 Jn (− jγx x) + C2 Yn (− jγx x)

(4.2.21)

where, Jn (− jγx x) is referred to as Bessel function of the first kind of order n, Yn (− jγx x) as the Bessel function of the second kind of order n. Finally, the solution of (4.2.15) becomes α1 f (x) = x n 1 C1 Jn 1 (− jγx x) + C2 Yn 1 (− jγx x) , n 1 = 1 − 2

(4.2.22)

Similarly, the solutions to (4.2.16) and (4.2.17) are obtained as α2 g(y) = y n 2 C3 Jn 2 (− jγ y y) + C4 Yn 2 (− jγ y y) , n 2 = 1 − 2 α3 n3 h(z) = z C5 Jn 3 (− jγz z) + C6 Yn 3 (− jγz z) , n 3 = 1 − 2

(4.2.23) (4.2.24)

From (4.2.14) and (4.2.22) through (4.2.24), the solution of (4.2.10) for E x (x, y, z) in D-dimensional fractional space have the form E x (x, y, z) = x n 1 y n 2 z n 3 C1 Jn 1 (− jγx x) + C2 Yn 1 (− jγx x) × C3 Jn 2 (− jγ y y) + C4 Yn 2 (− jγ y y) × C 5 Jn 3 (− jγz z) + C 6 Yn 3 (− jγz z) (4.2.25) where, C1 through C6 are constant coefficients.

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

Similarly, the solutions to (4.2.11) and (4.2.12) are obtained as E y (x, y, z) = x n 1 y n 2 z n 3 D1 Jn 1 (− jγx x) + D2 Yn 1 (− jγx x) × D3 Jn 2 (− jγ y y) + D4 Yn 2 (− jγ y y) × D5 Jn 3 (− jγz z) + D6 Yn 3 (− jγz z) and

E z (x, y, z) = x n 1 y n 2 z n 3 F1 Jn 1 (− jγx x) + F2 Yn 1 (− jγx x) × F3 Jn 2 (− jγ y y) + F4 Yn 2 (− jγ y y) × F5 Jn 3 (− jγz z) + F6 Yn 3 (− jγz z)

37

(4.2.26)

(4.2.27)

where, D1 through D6 and F1 through F6 are constant coefficients. For e jwt time variations, the instantaneous form E (x, y, z; t) of the vector complex function E(x, y, z) in (4.2.8) takes the form E (x, y, z; t) = e[{aˆ x E x (x, y, z) + aˆ y E y (x, y, z) +aˆ z E z (x, y, z)}e jwt ]

(4.2.28)

where, E x (x, y, z), E y (x, y, z) and E z (x, y, z) are given by (4.2.25) through (4.2.27). Equation (4.2.28) provides a general plane wave solution for lossy media in fractional space. This solution can be used to study the phenomenon of electromagnetic wave propagation in any non-integer dimensional space.

4.2.2 Discussion on Fractional Space Solution in Lossy Medium Equation (4.2.28) is the generalization of solution for Helmholtz’s equation in integer dimensional space to a non-integer dimensional space. As a special case, for threedimensional space, this problem reduces to classical wave propagation concept; i.e., if we set α1 = 1 in Eq. (4.2.22) then n 1 = 12 and it gives 1 f (x) = x 2 C 1 J 1 (− jγx x) + C2 Y 1 (− jγx x) (4.2.29) 2

2

Using Bessel functions of fractional order [4]: 2 sin (x) J 1 (x) = 2 πx 2 cos (x) Y 1 (x) = − 2 πx Eq. (4.2.22) can be reduced to

(4.2.30) (4.2.31)

38

4 Electromagnetic Wave Propagation in Fractional Space

f (x) =

2j C1 sinh(γx x) + C2 cosh(γx x) π γx

(4.2.32)

Similarly, we set α2 = 1 and α3 = 1 in (4.2.23) and (4.2.24) respectively and using Bessel function of fractional order in (4.2.30) through (4.2.31), we get

2j (4.2.33) C3 sinh(γ y y) + C4 cosh(γ y y) g(y) = π γy

h(z) =

2j C5 sinh(γz z) + C 6 cosh(γz z) π γz

(4.2.34)

From (4.2.32) through (4.2.34), we get E x (x, y, z) in D = 3 dimensional space as follows

8 C1 sinh(γx x) + C2 cosh(γx x) E x (x, y, z) = jπ 3 γx γ y γz × C 3 sinh(γ y y) + C4 cosh(γ y y) (4.2.35) × C5 sinh(γz z) + C 6 cosh(γz z) Similarly, for D = 3 Eqs. (4.2.26) and (4.2.27) can be reduced to

8 D1 sinh(γx x) + D2 cosh(γx x) E y (x, y, z) = 3 jπ γx γ y γz × D3 sinh(γ y y) + D4 cosh(γ y y) × D5 sinh(γz z) + D6 cosh(γz z)

E z (x, y, z) =

8 jπ 3 γ

x γ y γz

F1 sinh(γx x) + F2 cosh(γx x)

(4.2.36)

× F3 sinh(γ y y) + F4 cosh(γ y y) × F5 sinh(γz z) + F6 cosh(γz z)

(4.2.37)

Now, solution obtained in (4.2.35) through (4.2.37) is comparable to the well known solution of Helmholtz’s equation in lossy media present in integer dimensional space obtained by Balanis [1]. As a special case, if we choose a single parameter for non-integer dimension D where 2 < D ≤ 3, i.e., we take α1 = α2 = 1 so D = α3 + 2. In this case from Eq. (4.2.13) we obtain ∂2 Ex ∂2 Ex ∂2 Ex D − 3 ∂ Ex − γ 2 Ex = 0 + + + ∂x2 ∂ y2 ∂z 2 z ∂z

(4.2.38)

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

39

Solving this equation by separation of variables leads to the following result E x (x, y, z) = z n G 1 cosh(γx x) + G 2 sinh(γx x) × G 3 cosh(γ y y) + G 4 sinh(γ y y) (4.2.39) × G 5 Jn (− jγz z) + G 6 Yn (− jγz z) or E x (x, y, z) = z n G 1 cosh(γx x) + G 2 sinh(γx x) × G 3 cosh(γ y y) + G 4 sinh(γ y y) × G 5 Hn(1) (− jγz z) + G 6 Hn(2) (− jγz z)

(4.2.40)

In (4.2.39) and (4.2.40), n = 2 − D2 . Also G 1 through G 6 and G 5 through G 6 are constant coefficients. In (4.2.39) Jn (− jγz z) is referred to as Bessel function of the first kind of order n, Yn (− jγz z) as the Bessel function of the second kind of order n (1) and both are used to represent standing waves. In (4.2.40) Hn (− jγz z) is referred to (2) as Hankel function of the first kind of order n, Hn (− jγz z) as the Hankel function of the second kind of order n and both are used to represent traveling waves. Now, as a special case of fractional space solution, if we set D = 3 in (4.2.39), and use (4.2.30) and (4.2.31), we get

E x (x, y, z) =

2j G 1 cosh(γx x) + G 2 sinh(γx x) π γz × G 3 cosh(γ y y) + G 4 sinh(γ y y) × G 5 sinh(γz z) + G 6 cosh(γz z)

(4.2.41)

The result obtained in (4.2.41) is comparable to well know integer dimensional solution of wave equation obtained by Balanis [1]. Finally, if we take α = 0 in (4.2.4) then using γ = jβ in (4.2.25) through (4.2.27), we can reduce the fractional space solution for lossy medium to fractional space solution for lossless medium as below: E x (x, y, z) = x n 1 y n 2 z n 3 C1 Jn 1 (βx x) + C2 Yn 1 (βx x) × C 3 Jn 2 (β y y) + C4 Yn 2 (β y y) (4.2.42) × C5 Jn 3 (βz z) + C6 Yn 3 (βz z) E y (x, y, z) = x n 1 y n 2 z n 3 D1 Jn 1 (βx x) + D2 Yn 1 (βx x) × D3 Jn 2 (β y y) + D4 Yn 2 (β y y) × D5 Jn 3 (βz z) + D6 Yn 3 (βz z)

(4.2.43)

40

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.3 Current sheet as source of plane waves in fractional space

E z (x, y, z) = x n 1 y n 2 z n 3 F1 Jn 1 (βx x) + F2 Yn 1 (βx x) × F3 Jn 2 (β y y) + F4 Yn 2 (β y y) × F5 Jn 3 (βz z) + F6 Yn 3 (βz z)

(4.2.44)

Fractional space solution for lossless medium obtained in (4.2.42) through (4.2.44) is in exact agreement with that obtained in previous section.

4.2.3 Example: Current Sheet as Source of Plane Waves in Fractional Space As an example, an infinite sheet of surface current can be considered as a source of plane waves in lossy medium present in D-dimensional fractional space. We assume that an infinite sheet of electric surface current density Js = Js0 xˆ exists on the z = 0 plane in lossy medium characterized by ε1 , μ1 , σ1 in half-space z < 0 and by ε2 , μ2 , σ2 in half-space z > 0 (see Fig. 4.3). Since the sources do not vary with x or y, the fields will not vary with x or y but will propagate away from the source in ±z direction. The boundary conditions to be satisfied at z = 0 are zˆ × (E2 − E1 ) = 0 and zˆ × (H2 − H1 ) = Js0 x, ˆ where E1 , H1 are the fields for z < 0, and E2 , H2 are the fields for z > 0. To satisfy the later boundary condition, H must have a yˆ component. Then for E to be normal to H and zˆ , E must have an xˆ component. Thus, the corresponding wave equation for E and H fields in D-dimensional fractional space where 2 < D ≤ 3 can be written by modifying (4.2.38) as

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

41

Fig. 4.4 Usual wave propagation (D = 3) in lossy medium

d2 Ex D − 3 d Ex + − γ 2 Ex = 0 2 dz z dz d 2 Hy D − 3 d Hy + − γ 2 Hy = 0 2 dz z dz

(4.2.45) (4.2.46)

Solution of (4.2.45) and (4.2.46) takes the similar form as (4.2.40) and under above mentioned boundary conditions the fields will have the following form: Js0 n (1) z Hn (− jγ1 z); 2 Js H1 = yˆ 0 z n Hn(1) (− jγ1 z); 2η1 Js E2 = −xˆ 0 z n Hn(2) (− jγ2 z); 2 Js0 n (2) z Hn (− jγ2 z); H2 = − yˆ 2η2 E1 = −xˆ

z0

(4.2.50)

jωμ2 1 where, γ12 = jωμ1 (σ1 + jωε1 ), γ22 = jωμ2 (σ2 + jωε2 ), η1 = jωμ γ1 , η2 = γ2 . Assuming a time dependency e jwt and Js0 = −2 A/m, the solution for the usual wave for z > 0 with D = 3 is shown in Fig. 4.4, which is comparable to well known plane wave solutions for 3-dimensional space in [1]. Similarly, for D = 2.5 we have fractal wave propagating in fractional space for z > 0 as shown in Fig. 4.5, where amplitude variations are described in terms of Hankel functions.

42

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.5 Wave propagation in lossy medium present in fractional space (D = 2.5)

4.2.4 Summary The phenomenon of wave propagation in source-free and lossy media in fractional space is studied by solving Helmholtz’s equation in D-dimensional fractional space. When the wave propagates in lossy media in the fractional space, the amplitude variations are described by Bessel functions. For all investigated cases when D is an integer-dimension, the classical results are recovered. The plane wave solutions investigated in this paper have potential applications in wave propagation problems in fractional space.

4.3 Cylindrical Wave Propagation in Fractional Space In this section, we present an exact solution of cylindrical wave equation in fractional space that can be used to describe the phenomenon of wave propagation in any fractal media. In Sect. 4.3.1, we investigate full analytical cylindrical wave solution to the wave equation in D-dimensional fractional space, where the parameter D is used to describe the measure distribution of space. In Sect. 4.3.2, the solution of wave equation in integer-dimensional space is justified from the results of previous section. Finally, in Sect. 4.3.3, major results are summarized.

4.3 Cylindrical Wave Propagation in Fractional Space

43

4.3.1 An Exact Solution of Cylindrical Wave Equation in Fractional Space The problems that exhibit cylindrical geometries are needed to be solved using cylindrical coordinate system. As for the case of rectangular geometries, the electric and magnetic fields of cylindrical geometry boundary-value problem must satisfy corresponding cylindrical wave equation [1]. Let us assume that the space in which fields must be solved is fractional dimensional and source-free. For source-free and lossless media, the vector wave equations for the complex electric and magnetic field intensities are given by the Helmholtz’s equation as follows [1]. 2 ∇D E + β2E = 0

(4.3.1)

2 H + β2H = 0 ∇D

(4.3.2)

where, β 2 = ω2 με. Time dependency e jwt has been suppressed throughout the 2 is the Laplacian operator in D-dimensional fractional space discussion. Here, ∇ D and is defined in rectangular coordinate system as follows [2]. 2 ∇D =

∂2 α1 − 1 ∂ α2 − 1 ∂ α3 − 1 ∂ ∂2 ∂2 + + + + + 2 2 2 ∂x x ∂x ∂y y ∂y ∂z z ∂z

(4.3.3)

where x, y and z are rectangular coordinates. Equation (4.3.3) uses three parameters (0 < α1 ≤ 1, 0 < α2 ≤ 1 and 0 < α3 ≤ 1) to describe the measure distribution of space where each one is acting independently on a single coordinate and the total dimension of the system is D = α1 + α2 + α3 . To find cylindrical wave solutions of wave equation in D-dimensional fractional space, it is likely that a cylindrical coordinate system (ρ, φ, z) will be used. In cylindrical coordinate system (4.3.3) becomes 2 ∇D =

∂2 1 ∂ + (α1 + α2 − 1) ∂ρ 2 ρ ∂ρ 2 1 ∂ ∂ + 2 − {(α1 − 1) tan φ − (α2 − 1) cot φ} ρ ∂φ 2 ∂φ +

∂2 α3 − 1 ∂ + 2 ∂z z ∂z

(4.3.4)

Once the solution to any one of Eqs. (4.3.1) and (4.3.2) in fractional space is known, the solution to the other can be written by an interchange of E with H or H with E due to duality [1]. We will examine the solution for E. In cylindrical coordinates, a general solution for E can be written as E(ρ, φ, z) = aˆ ρ E ρ (ρ, φ, z) + aˆ φ E φ (ρ, φ, z) + aˆ z E z (ρ, φ, z) Substituting (4.3.5) into (4.3.1) we can write that

(4.3.5)

44

4 Electromagnetic Wave Propagation in Fractional Space 2 ∇D (aˆ ρ E ρ + aˆ φ E φ + aˆ z E z ) + β 2 (aˆ ρ E ρ + aˆ φ E φ + aˆ z E z ) = 0

(4.3.6)

Since, 2 2 (aˆ ρ E ρ ) = aˆ ρ ∇ D Eρ ∇D

(4.3.7)

2 2 ∇D (aˆ φ E φ ) = aˆ φ ∇ D Eφ

(4.3.8)

2 2 (aˆ z E z ) = aˆ z ∇ D Ez ∇D

(4.3.9)

So, Eq. (4.3.6) cannot be reduced to simple scalar wave equations, but it can be reduced to coupled scalar partial differential equations. However for simplicity, the wave mode solution can be formed in cylindrical coordinates that must satisfy the following scalar wave equation: 2 ψ(ρ, φ, z) + β 2 ψ(ρ, φ, z) = 0 ∇D

(4.3.10)

where, ψ(ρ, φ, z) is a scalar function that can represent a field or vector potential component. In expanded form (4.3.10) can be written as ∂ 2ψ 1 ∂ψ + (α1 + α2 − 1) ∂ρ 2 ρ ∂ρ 2 1 ∂ψ ∂ψ + 2 − {(α − 1) tan φ − (α − 1) cot φ} 1 2 ρ ∂φ 2 ∂φ +

∂ 2ψ α3 − 1 ∂ψ + + β 2ψ = 0 2 ∂z z ∂z

(4.3.11)

Equation (4.3.11) is separable using method of separation of variables. We consider ψ(ρ, φ, z) = f (ρ)g(φ)h(z)

(4.3.12)

the resulting ordinary differential equations are obtained as follows: 2 d2 d ρ 2 2 + ρ(α1 + α2 − 1) + βρ ρ − m 2 f (ρ) = 0 dρ dρ 2 d d 2 + {(α1 − 1) tan φ + (α2 − 1) cot φ} − m g(φ) = 0 dφ 2 dφ 2 α3 − 1 d d 2 + β + z h(z) = 0 dz 2 z dz where, m is a constant (integer usually). In addition,

(4.3.13) (4.3.14) (4.3.15)

4.3 Cylindrical Wave Propagation in Fractional Space

45

βρ2 + βz2 = β 2

(4.3.16)

Equation (4.3.16) is referred to as constraint equation. In addition βρ , βz are known as wave constants in the ρ, z directions, respectively, which will be determined using boundary conditions. Now, Eq. (4.3.13) through (4.3.15) are needed to be solved for f (ρ), g(φ) and h(z), respectively. We choose to work first with f (ρ). Equation (4.3.13) can be written as: 2 d 2 d + aρ (4.3.17) + bρ + c f (ρ) = 0 ρ dρ 2 dρ where, a = α1 + α2 − 1, b = βρ2 , c = −m 2 , = 2. Equation (4.3.15) is closely related to Bessel’s equation and its solutions is given as [3]: 1−a 2√ 2√ 2 2 2 C1 Jv ( bρ ) + C2 Yv ( bρ ) (4.3.18) f (ρ) = ρ where, v = 1 (1 − a)2 − 4c Using (4.3.18), the final solution of (4.3.13) is given by f 1 (ρ) = ρ 1− or f 2 (ρ) = ρ 1−

α1 +α2 2

α1 +α2 2

C1 Jv (βρ ρ) + C2 Yv (βρ ρ)

D1 Hv(1) (βρ ρ) + D2 Hv(2) (βρ ρ)

(4.3.19)

(4.3.20)

where, v = 12 (2 − α1 − α2 )2 + 4m 2 . In (4.3.19) Jv (βρ ρ) is referred to as Bessel function of the first kind of order v and Yv (βr ) as the Bessel function of the second kind of order v. They are used to represent standing waves. In (4.3.20) Hv(1) (βr ) (2) is referred to as Hankel function of the first kind of order v and Hv (βρ ρ) as the Hankel function of the second kind of order v, and are used to represent traveling waves. Now, we find the solution of equation (4.3.14) for g(φ). Equation (4.3.14) can be reduced to following Gaussian hypergeometric equation after proper mathematical steps under substitution ξ = sin2 (φ) [3]: ξ(1 − ξ )

dg(φ) d 2 g(φ) + {(A + B + 1)ξ − C} + ABg(φ) = 0 2 dξ dξ

(4.3.21)

where, 1 (2 − α2 + α1 ) 2 m2 AB = − 4

A+ B+1=

(4.3.22) (4.3.23)

46

4 Electromagnetic Wave Propagation in Fractional Space

C=

1 (2 − α2 ) 2

(4.3.24)

solution to equation (4.3.21) is given as [3]: g(φ) = C3 F(A, B, C; ξ ) + C4 ξ 1−C F(A − C + 1, B − C + 1, 2 − C; ξ ) (4.3.25) where, F(A, B, C; ξ ) = 1 +

∞ (A)k (B)k ξ k (C)k k!

(4.3.26)

k=1

with, (A)k = A(A + 1) . . . (A + k + 1)

(4.3.27)

F(A, B, C; ξ ) is known as Gaussian hypergeometric function, and A, B, C are known from (4.3.22) through (4.3.24). Now, we find the solution of equation (4.3.15) for h(z). Equation (4.3.15) can be written as: 2 d d 2 (4.3.28) + βz z h(z) = 0 z 2 +e dz dz where, e = α3 − 1. Equation (4.3.28) is reducible to Bessel’s equation under substitution h = z n ζ as follows: d2 d |1 − e| z2 2 + z + (βz2 z 2 − n 2 ) ζ (z) = 0, n = (4.3.29) dz dz 2 The solution of Bessel’s equation in (4.3.29) is given as [3] ζ (z) = C5 Jn (βz z) + C 6 Yn (βz z)

(4.3.30)

where, Jn (βz z) is referred to as Bessel function of the first kind of order n, Yn (βz z) as the Bessel function of the second kind of order n. Finally the solution of (4.3.15) becomes α3 (4.3.31) h(z) = z n C5 Jn (βz z) + C6 Yn (βz z) , n = 1 − 2 The appropriate solution forms of f (ρ), g(φ) and h(z) depend upon the problem. From (4.3.12), (4.3.19), (4.3.25) and (4.3.31), a typical solution for ψ(r, θ, φ) to represent the fields within a cylindrical geometry may take the form ψ(ρ, φ, z) = [ρ 1−

α1 +α2 2

+ C4 ξ

{C 1 Jv (βρ ρ) + C2 Yv (βρ ρ)}] × [{C3 F(A, B, C; ξ )

F(A − C + 1, B − C + 1, 2 − C; ξ )}] × [z {C 5 Jn (βz z) + C 6 Yn (βz z)}] n

1−C

(4.3.32)

4.3 Cylindrical Wave Propagation in Fractional Space

47

where, ξ = sin2 (φ) and C1 through C6 are constant coefficients. Equation (4.3.32) provides a general solution to cylindrical wave equation in fractional space. This solution can be used to study the phenomenon of electromagnetic wave propagation in any non-integer dimensional space.

4.3.2 Discussion on Cylindrical Wave Solution in Fractional Space Equation (4.3.32) is the generalization of the concept of wave propagation from integer dimensional space to the non-integer dimensional space. As a special case, for three-dimensional space, this problem reduces to classical wave propagation concept; i.e., as a special case, if we set α1 = α2 = α3 = 1 in Eqs. (4.3.19), (4.3.25) and (4.3.31), we get cylindrical wave solution in integer dimensional space. For α1 = α2 = 1 Eqs. (4.3.19) and (4.3.20) provide f 1 (ρ) = C 1 Jm (βρ ρ) + C 2 Ym (βρ ρ)

(4.3.33)

f 2 (ρ) = D1 Hm(1) (βρ ρ) + D2 Hm(2) (βρ ρ)

(4.3.34)

and

Similarly, if we set α1 = α2 = 1 in Eqs. (4.3.22) and (4.3.24), we get A = −B = C = 21 . Now, considering following special forms of Gaussian hypergeometric function [4]: m 2,

1 F(λ, −λ, ; sin2 v) = cos(2λv) 2 sin[(2λ − 1)v] 3 F(λ, 1 − λ, ; sin2 v) = 2 (2λ − 1) sin(v)

(4.3.35) (4.3.36)

Eq. (4.3.25) can be reduced to g(φ) = C3 cos(mφ) + C4 sin(mφ) In a similar way, if we set α3 = 1 in (4.3.32) then n = 12 and it gives 1 h(z) = z 2 C 5 J 1 (βz z) + C6 Y 1 (βz z) 2

2

Using Bessel functions of fractional order [4]: 2 sin (z) J 1 (z) = 2 πz 2 cos (z) Y 1 (z) = − 2 πz

(4.3.37)

(4.3.38)

(4.3.39) (4.3.40)

48

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.6 Cylindrical waveguide of circular cross section.[This figure was originally published in [6] , reproduced courtesy of The Electromagnetics Academy]

equation (4.3.13) can be reduced to h(z) = C5 sin(βz z) + C6 cos(βz z)

(4.3.41)

where, Ci = Ci πβ2 z , i = 5, 6 From (4.3.12), (4.3.33), (4.3.37) and (4.3.41), a typical solution in three dimensional space ( a special case of fractional space) for ψ(ρ, φ, z) to represent the fields within a cylindrical geometry will take the form ψ(ρ, φ, z) = C1 Jm (βρ ρ) + C2 Ym (βρ ρ) × [C3 cos(mφ) + C4 sin(mφ)] × [C5 sin(βz z) + C6 cos(βz z)] (4.3.42) which is comparable to the cylindrical wave solutions of the wave equation in integer dimensional space obtained by Balanis [1]. As an example, the fields inside a circular waveguide filled with fractal media of dimension D can be obtained by assuming a D-dimensional fractional space inside the circular waveguide. Within such circular waveguide of radius a (see Fig. 4.6), standing waves are created in the radial(ρ) direction, periodic waves in the φ-direction, and traveling waves in the z-direction. For the fields to be finite at ρ = 0 where Y v(βρ ρ) possesses a singularity, (4.3.32) reduces to ψ1 (ρ, φ, z) =[ρ 1−

α1 +α2 2

+ C4 ξ

{C1 Jv (βρ ρ)}] × [{C3 F(A, B, C; ξ )

1−C

F(A − C + 1, B − C + 1, 2 − C; ξ )}]

× [z n {C5 Hn(2) (βz z) + C6 Hn(1) (βz z)}]

(4.3.43)

4.3 Cylindrical Wave Propagation in Fractional Space

49

Fig. 4.7 Cylindrical wave propagation in Euclidean space (D = 3). [This figure was originally published in [6] , reproduced courtesy of The Electromagnetics Academy]

Fig. 4.8 Cylindrical wave propagation in fractional space (D = 2.5). [ This figure was originally published in [6] , reproduced courtesy of The Electromagnetics Academy]

To represent the fields in the region outside the cylinder, where three dimensional space is assumed because there is no fractal media outside the cylinder, a typical solution for ψ(ρ, φ, z) would take the form

50

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.9 Cylindrical wave propagation in fractional space (D = 2.1). [ This figure was originally published in [6] , reproduced courtesy of The Electromagnetics Academy]

ψ2 (ρ, φ, z) = C2 Hm(2) (βρ ρ) × [C3 cos(mφ) + C4 sin(mφ)] × [C5 sin(βz z) + C 6 cos(βz z)]

(4.3.44)

In the region outside the cylinder, outward traveling waves are formed, in contract to standing waves inside the cylinder. In this way, the general cylindrical wave solution in fractional space can be used to study the wave propagation in the cylindrical geometries containing fractal media. Now, as another example we assume that a cylindrical wave exists in a fractional space due to some infinite line source. Since the source do not vary with z, the fields will not vary with z but will propagate away from the source in ρ-direction. Also for simplicity, we choose to visualize only the radial amplitude variations of scalar field ψ in fractional space which is given by (4.3.32) as: ψ(ρ) = Aρ 1−

α1 +α2 2

Hv(2) (βρ ρ)

(4.3.45)

Also, if we choose a single parameter for non-integer dimension D where 2 < D ≤ 3, i.e, we take α2 = α3 = 1 so D = α1 + 2. In this case (4.3.45) becomes ψ(ρ) = Aρ

3−D 2

Hv(2) (βρ ρ)

(4.3.46)

In (4.3.46), using asymptotic expansions of Hankel functions [4] for ρ → ∞, we see that the amplitude variations of field ψ are related with radial distance ρ as D

ψ(ρ) ∝ ρ 1− 2 From (4.3.47),

(4.3.47)

4.3 Cylindrical Wave Propagation in Fractional Space

for D = 3,ψ(ρ) ∝

51

√1 ρ

for D = 2.5, ψ(ρ) ∝ for D = 2.1, ψ(ρ) ∝

1 ρ 0.25 1 ρ 0.05

Assuming a time dependency e jwt , the radial amplitude variations of scalar field ψ are shown for different values of dimension D in Figs 4.7, 4.8, 4.9. It is seen that the amplitude of cylindrical wave propagating in higher dimensional space decays rapidly.

4.3.3 Summary An exact solution of cylindrical wave equation for electromagnetic field in D-dimensional fractional space is presented. The obtained exact solution of cylindrical wave equation is a generalization of classical integer-dimensional solution to a non-integer dimensional space. For all investigated cases when D is an integer dimension, the classical results are recovered. The investigated solution provides a basis for the application of the concept of fractional space to the wave propagation phenomenon in fractal media.

4.4 Spherical Wave Propagation in Fractional Space In this section we provide an exact solution of the spherical wave equation in D-dimensional fractional space which describes the phenomenon of electromagnetic wave propagation in fractional space. In Sect. 4.4.1, we investigate full analytical solution of spherical wave equation in D-dimensional fractional space, where a parameter D is used to describe the measure distribution of space. In Sect. 4.4.2, solution of wave equation in integer-dimensional space is justified from the results of previous section. Finally, in Sect. 4.4.3, results are summarized.

4.4.1 Spherical Wave Equation in D-dimensional Fractional Space The problems that exhibit spherical geometries are needed to be solved using spherical coordinates. As for the case of rectangular geometries, the electric and magnetic fields of spherical geometry boundary-value problem must satisfy corresponding spherical wave equation [1]. Let us assume that the space in which fields must be solved is fractional dimensional, source-free and lossless. For source-free and lossless media, the vector wave equations for the complex electric and magnetic field intensities are given by the Helmholtz’s equation as follows:

52

4 Electromagnetic Wave Propagation in Fractional Space 2 ∇D E + β2E = 0

(4.4.1)

2 H + β2H = 0 ∇D

(4.4.2)

where, β 2 = ω2 με. Time dependency e jwt has been suppressed throughout the 2 is the Laplacian operator in D-dimensional fractional space discussion. Here ∇ D and is defined in spherical coordinates as follows [2]: 2 ∂2 D−1 ∂ D−2 ∂ 1 ∂ 2 = 2+ + + 2 ∇D ∂r r ∂r r ∂θ 2 tan θ ∂θ 2 1 D−3 ∂ ∂ + 2 + r sin θ ∂φ 2 tan φ ∂φ (4.4.3) where, 2 < D ≤ 3. Once the solution to any one of Eq. (4.4.1) and (4.4.2) in fractional space is known, the solution to the other can be written by an interchange of E with H or H with E due to duality. We will examine the solution for E. In spherical coordinates, a general solution for E can be written as E(r, θ, φ) = aˆr Er (r, θ, φ) + aˆ θ E θ (r, θ, φ) + aˆ φ E φ (r, θ, φ)

(4.4.4)

Substituting (4.4.4) into (4.4.1) we can write that 2 (aˆ r E r + aˆ θ E θ + aˆ φ E φ ) + β 2 (aˆ r Er + aˆ θ E θ + aˆ φ E φ ) = 0 ∇D

(4.4.5)

Since, 2 2 (aˆ r Er ) = aˆ r ∇ D Er ∇D

(4.4.6)

2 2 ∇D (aˆ θ E θ ) = aˆ θ ∇ D Eθ

(4.4.7)

2 2 (aˆ φ E φ ) = aˆ φ ∇ D Eφ ∇D

(4.4.8)

So, Eq. (4.4.5) cannot be reduced to three simple scalar wave equations, but it can be reduced to three coupled scalar partial differential equations. However for simplicity, the wave mode solution can be formed in spherical coordinates that must satisfy the following scalar wave equation: 2 ψ(r, θ, φ) + β 2 ψ(r, θ, φ) = 0 ∇D

(4.4.9)

where, ψ(r, θ, φ) is a scalar function that can represent a field or vector potential component. In expanded form (4.4.9) can be written as 1 ∂ 2ψ D − 1 ∂ψ D − 2 ∂ψ ∂2ψ + 2 + + ∂r 2 r ∂r r ∂θ 2 tan θ ∂θ 2 1 D − 3 ∂ψ ∂ ψ + 2 + + β 2ψ = 0 r sin θ ∂φ 2 tan φ ∂φ (4.4.10)

4.4 Spherical Wave Propagation in Fractional Space

53

Equation (4.4.10) is separable using method of separation of variables. We consider ψ(r, θ, φ) = f (r )g(θ )h(φ)

(4.4.11)

the resulting ordinary differential equations are obtained as follows: 2 d 2 d 2 + (D − 1)r + (βr ) − n(n + 1) f = 0 r dr 2 dr

m 2 D−2 d d2 + + n(n + 1) g=0 − dθ 2 tan θ dθ sin θ 2 D−3 d d 2 + + m h=0 dφ 2 tan φ dφ

(4.4.12)

(4.4.13) (4.4.14)

where, m and n are constants (integers usually). Now, Eq. (4.4.12) through (4.4.14) are needed to be solved for f (r ), g(θ ) and h(φ), respectively. For simplicity we write f (r ), g(θ ), h(φ) as f, g, h, respectively, inside all equations. We choose to work first with f (r ). Equation (4.4.12) can be written as: 2 s d 2 d + ar (4.4.15) + br + c f = 0 r dr 2 dr where, a = D − 1, b = β 2 , c = −n(n + 1), s = 2. Equation (4.4.15) is closely related to Bessel’s equation and its solutions is given as [3]: 1−a 2√ s 2√ s f = r 2 C1 Jv ( br 2 ) + C2 Yv ( br 2 ) (4.4.16) s s where, v = 1s (1 − a)2 − 4c Using (4.4.16), the final solution of (4.4.12) is given by D

f 1 = r 1− 2 [C1 Jv (βr ) + C2 Yv (βr )]

(4.4.17)

D f 2 = r 1− 2 D1 Hv(1) (βr ) + D2 Hv(2) (βr )

(4.4.18)

or

where, v = 12 (2 − D)2 + 4n(n + 1). In (4.4.17) Jv (βr ) is referred to as Bessel function of the first kind of order v and Yv (βr ) as the Bessel function of the second kind of order v. They are used to represent radial standing waves. In (4.4.18) Hv(1) (βr ) (2) is referred to as Hankel function of the first kind of order v and Hv (βr ) as the Hankel function of the second kind of order v, and are used to represent radial traveling waves.

54

4 Electromagnetic Wave Propagation in Fractional Space

Now, we find the solution of Eq. (4.4.13) for g(θ ). Equation (4.4.13) can be reduced to following Gaussian hypergeometric equation after proper mathematical : steps under substitution g = w sinm θ and z 1 = 1+cosθ 2 z 1 (1 − z 1 )

d2w dw + {(α1 + β1 + 1)z 1 − γ1 } + α1 β1 w = 0 2 dz 1 dz 1

(4.4.19)

where, α1 + β1 + 1 = −2(D − 2)(n + 1)

(4.4.20)

α1 β1 = (n − m)(m + n − 1)

(4.4.21)

γ1 = −(D − 2)(n + 1)

(4.4.22)

solution to Eq. (4.4.19) is given as [3]: w = C3 F(α1 , β1 , γ1 ; z 1 ) 1−γ1

+ C4 z 1

F(α1 − γ1 + 1, β1 − γ1 + 1, 2 − γ1 ; z 1 )

(4.4.23)

where, F(α1 , β1 , γ1 ; z 1 ) = 1 +

∞ (α1 )k (β1 )k z 1k (γ1 )k k!

(4.4.24)

k=1

with, (α1 )k = α1 (α1 + 1)...(α1 + k + 1)

(4.4.25)

F(α, β, γ ; z) is known as Gaussian hypergeometric function, and α1 , β1 , γ1 are known from (4.4.20) through (4.4.22). From (4.4.19) through (4.4.23), the final solution of (4.4.13) is given by g = [C 3 F(α1 , β1 , γ1 ; z 1 ) 1−γ1

+ C4 z 1

F(α1 − γ1 + 1, β1 − γ1 + 1, 2 − γ1 ; z 1 )]sin m θ

(4.4.26)

Now, we find the solution of equation (4.4.14) for h(φ). Equation (4.4.14) can be written as: 2 d d +q h =0 (4.4.27) + pcotφ dφ 2 dφ

4.4 Spherical Wave Propagation in Fractional Space

55

where, p = D − 3, and q = m 2 . Equation (4.4.27) can be reduced to following Gaussian hypergeometric equation after proper mathematical steps under substitution : z 2 = 1+cosφ 2 z 2 (1 − z 2 )

d 2h dh + {(α2 + β2 + 1)z 2 − γ2 } + α2 β2 h = 0 2 dz 2 dz 2

(4.4.28)

where, α2 + β2 = D − 5

(4.4.29)

α2 β2 = m 2

(4.4.30)

γ2 =

1 (4 − D) 2

(4.4.31)

From (4.4.28) through (4.4.31), the final solution of (4.1.4) is given as h = C5 F(α2 , β2 , γ2 ; z 2 ) 1−γ2

+ C6 z 2

F(α2 − γ2 + 1, β2 − γ2 + 1, 2 − γ2 ; z 2 )

(4.4.32)

where, F is Gaussian hypergeometric function described in (4.2.4) through (4.2.5), and α2 , β2 , γ2 are known from (4.2.9) through (4.3.1). The appropriate solution forms of f, g and h depend upon the problem geometry. From (4.4.11), (4.1.7), (4.4.26) and (4.4.32), a typical solution for ψ(r, θ, φ) to represent the fields within a spherical geometry may take the form D

ψ(r, θ, φ) = [r 1− 2 {C 1 Jv (βr ) + C2 Yv (βr )}] × [{C3 F(α1 , β1 , γ1 ; z 1 ) 1−γ

+ C 4 z 1 1 F(α1 − γ1 + 1, β1 − γ1 + 1, 2 − γ1 ; z 1 )}sin m θ ] × [C 5 F(α2 , β2 , γ2 ; z 2 ) 1−γ2

+ C6 z 2

F(α2 − γ2 + 1, β2 − γ2 + 1, 2 − γ2 ; z 2 )]

(4.4.33) 1+cosφ , z = and C through C are constant coefficients. where, z 1 = 1+cosθ 2 1 6 2 2 Equation (4.4.33) provides a general solution to spherical wave equation in fractional space. This solution can be used to study the phenomenon of electromagnetic wave propagation in a non-integer dimensional space.

4.4.2 Discussion on Fractional Space Solution Equation (4.4.33) is the generalization of the concept of wave propagation in integer dimensional space to the wave propagation in non-integer dimensional space. As

56

4 Electromagnetic Wave Propagation in Fractional Space

a special case, for three-dimensional space, this problem reduces to classical wave propagation concept; i.e., as a special case, if we set D = 3 in Eq. (4.4.17) then v = n + 21 and it gives 1 f 1 = √ C1 Jn+ 1 (βr ) + C2 Yn+ 1 (βr ) 2 2 r

(4.4.34)

here, regular Bessel functions in (4.4.34) are related to spherical Bessel function by [4] π jn (βr ) = (4.4.35) J 1 (βr ) 2βr n+ 2 π Y 1 (βr ) yn (βr ) = (4.4.36) 2βr n+ 2 from (4.4.35) and (4.4.36) we can reduce (4.3.4) to 2β f1 = [C1 jn (βr ) + C2 yn (βr )] π similarly, for three dimensional space we reduce (4.1.8) to 1 (1) (2) f 2 = √ C 1 H 1 (βr ) + C2 H 1 (βr ) n+ 2 n+ 2 r

(4.4.37)

(4.4.38)

here, regular Hankel functions in (4.4.38) are related to spherical Hankel function by [4] π (1) (1) h n (βr ) = (4.4.39) H 1 (βr ) 2βr n+ 2 π (2) (4.4.40) h (2) H 1 (βr ) n (βr ) = 2βr n+ 2 from (4.4.39) and (4.4.40), we can reduce (4.3.8) to 2β (2) D1 h (1) f2 = n (βr ) + D2 h n (βr ) π

(4.4.41)

Similarly, we set D = 3 in (4.2.26) and solving (4.1.19) for this special case, we get d m Pn (ξ ) d m Q n (ξ ) g = C3 sin m θ + C4 (4.4.42) dξ m dξ m where, ξ = cos θ , Pn (ξ ) and Q n (ξ ) are referred to as Legendre functions of first and second kind and are related to associated Legendre functions Pnm (ξ ) and Q m n (ξ ) as [4]

4.4 Spherical Wave Propagation in Fractional Space

57 m

Pnm (ξ ) = (1 − ξ 2 ) 2 m

2 2 Qm n (ξ ) = (1 − ξ )

d m Pn (ξ ) dξ m

(4.4.43)

d m Q n (ξ ) dξ m

(4.4.44)

From (4.4.42) through (4.4.44), we get g in three-dimensional space as : g = C3 Pnm (cos θ ) + C4 Q m n (cos θ )

(4.4.45)

In a similar way, we set D = 3 in (4.3.2) and using special forms of Gaussian hypergeometric functions [4], we get h in three dimensional space as: h = C5 cos(mφ) + C 6 sin(mφ)

(4.4.46)

From (4.4.11), (4.4.37), (4.4.45) and (4.4.46), a typical solution in three dimensional space ( a special case of fractional space) for ψ(r, θ, φ) to represent the fields within a spherical geometry will take the form ψ(r, θ, φ) =

2β [C1 jn (βr ) + C2 yn (βr )] π × [C3 Pnm (cos θ ) + C4 Q m n (cos θ )] × [C5 cos(mφ) + C6 sin(mφ)]

(4.4.47)

which is comparable to the solution of spherical wave equation in integer dimensional space obtained by Balanis [1]. Now, as an example we assume that a spherical wave exists in a fractional space due to some point source. For simplicity, we choose to visualize only the radial amplitude variations of scalar field ψ in fractional space which is given by (4.1.8) as: ψ(r ) = Ar 1− 2 Hv(2) (βr ) D

(4.4.48)

where, 2 < D ≤ 3. In (4.4.48), using asymptotic expansions of Hankel functions [1] for r → ∞, we see that the amplitude variations of field ψ are related with radial distance r as ψ(r ) ∝ r From (4.4.49), for D = 3,ψ(r ) ∝ r1 1 for D = 2.5, ψ(r ) ∝ r 0.75 1 for D = 2.1,ψ(r ) ∝ r 0.55

1−D 2

(4.4.49)

58

4 Electromagnetic Wave Propagation in Fractional Space

Fig. 4.10 Spherical wave propagation in Euclidean space (D = 3). [ This figure was originally published in [7] , reproduced courtesy of The BRILL]

Fig. 4.11 Spherical wave propagation in fractional space (D = 2.5). [ This figure was originally published in [7] , reproduced courtesy of The BRILL]

Assuming a time dependency e jwt , the radial amplitude variations of scalar field ψ are shown for different values of dimension D in Figs. 4.10, 4.11, 4.12. In Fig. 4.10 a spherical wave propagating in Euclidean space of dimension D = 3 is shown, where the amplitude decays by 1r . In Figs. 4.3, 4.4, 4.5 a spherical wave propagating

4.4 Spherical Wave Propagation in Fractional Space

59

Fig. 4.12 Spherical wave propagation in fractional space (D = 2.1). [ This figure was originally published in [7] , reproduced courtesy of The BRILL]

in fractional space of dimension D = 2.5 is shown, where the amplitude decays 1 by r 0.75 . In Fig. 4.12 a spherical wave propagating in fractional space of dimension 1 D = 2.1 is shown, where the amplitude decays by r 0.55 . It is seen that the amplitude of spherical wave propagating in higher dimensional space decays rapidly as compared to another spherical wave propagating in relatively lower dimensional space.

4.4.3 Summary An exact solution of the spherical wave equation is obtained in D-dimensional fractional space. The obtained fractional space solution provides a generalization of electromagnetic wave propagation phenomenon from integer space to fractional space. For integer values of dimension D, the classical results are recovered from fractional solution. The presented fractional space solution of the wave equation can be used to describe the phenomenon of wave propagation in any fractal media.

References 1. C.A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, 1989) 2. C. Palmer, P.N. Stavrinou, Equations of motion in a noninteger-dimension space. J. Phys. A 37, 6987–7003 (2004) 3. A.D. Polyanin, V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. (CRC Press, Boca Raton, 2003)

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4 Electromagnetic Wave Propagation in Fractional Space

4. M. Abramowitz, I.A. Stegun, Handbook of mathematical functions with formulas, Graphs, and Mathematical Tables.U.S. Department of Commerce (1972) 5. M. Zubair, M.J. Mughal, Q.A. Naqvi, The wave equation and general plane wave solutions in fractional space. Prog Electromn Res Lett 19, 137–146 (2010) 6. M. Zubair, M.J. Mughal, Q.A. Naqvi, An exact solution of cylindrical wave equation for electromagnetic field in fractional dimensional space. Prog Electromn Res Lett 114, 443–455 (2011) 7. M. Zubair, M.J. Mughal, Q.A. Naqvi, An exact solution of spherical wave in D-dimensional fractional space. J Electromn Waves Appl 25, 1481–1491 (2011)

Chapter 5

Electromagnetic Radiations from Sources in Fractional Space

In this chapter we present a procedure for solution of antenna radiation problems in fractional space along with an application of this novel procedure to the case of Hertzian dipole in fractional space. In Sect. 5.1, a novel solution procedure for antenna radiation problems in fractional space is proposed. In Sect. 5.2, the reported solution procedure is applied to the case of Hertzian dipole in fractional space. And finally major results are summarized in Sect. 5.3

5.1 Solution Procedure for Radiation Problems in Fractional Space In analysis of radiation problems, the procedure is to specify sources and get the fields radiated by the sources. A common practice in analysis of radiation problems in fractional space is to introduce auxiliary potential function AD (magnetic vector potential) and FD (electric vector potential) [1]. An overview of steps involved in solving typical radiation problems in fractional space are shown in Fig. 5.1 .

5.1.1 The Vector Potential AD for Electric Current Source J The vector potential AD in D-dimensional fractional space is useful in solving for the electromagnetic field generated by a given harmonic electric current J [1]. The fractional space generalization of the relation between AD and J is given by vector potential wave equation as below: 2 ∇D AD + k 2 AD = −μJ

(5.1.1)

2 is the laplacian operator in D-dimensional fractional space given by where, ∇ D Eq. 3.2.3, and k 2 = ω2 με.

M. Zubair et al., Electromagnetic Fields and Waves in Fractional Dimensional Space, SpringerBriefs in Applied Sciences and Technology, DOI: 10.1007/978-3-642-25358-4_5, © The Author(s) 2012

61

62

5 Electromagnetic Radiations from Sources in Fractional Space

Fig. 5.1 Block diagram for computing radiated fields in fractional space

Using the solution of Poisson’s equation in fractional space [2] and considering analogy of (5.1.1) with vector wave equation in fractional space solved in Sect. 4.1, we solve (5.1.1) for AD and get AD =

μ 23−D Γ (3/2) 4π Γ (D/2)

J(x , y , z )

(2)

Hn (k R) dV R D−5/2

(5.1.2)

where, the primed coordinates (x , y , z ) represent the location of source, n = 1− D2 , R is the distance between any point in the source to observation point, Γ (x) is the (2) gamma function and Hn (k R) denotes the Hankel function of second kind of order n representing outward going waves from source point. Now, to validate our provided solution in (5.1.2), we get vector potential A from our solution by substituting D = 3 in (5.1.2). For D = 3, we have n = −1/2. Using following Hankel function of fractional order [3] 2 −jx 2 (5.1.3) e H 1 (x) = πx 2 Equation 5.1.2 gets reduced to AD = C

μ 4π

J(x , y , z )

e− jk R dV R

(5.1.4)

where, C is a constant term. Equation 5.1.4 is in exact agreement with the solution provided in [1] for Euclidean space.

5.1.2 The Vector Potential FD for Magnetic Current Source M Although magnetic currents appear to be physically unrealizable, equivalent magnetic currents are considered when we use surface or volume equivalence theorems [4]. Similar to previous case, the vector potential FD in D-dimensional fractional space is useful in solving for the electromagnetic field generated by a given harmonic electric current H [1]. The fractional space generalization of the relation between FD and M is given by vector potential wave equation as follows: 2 FD + k 2 FD = −εM ∇D

(5.1.5)

5.1 Solution Procedure for Radiation Problems in Fractional Space

63

2 is the laplacian operator in D-dimensional fractional space given by where, ∇ D (3.2.3), and k 2 = ω2 με. Using the analogy to the solution provided for AD in (5.1.2), we get

ε 23−D Γ (3/2) FD = 4π Γ (D/2)

M(x , y , z )

(2)

Hn (k R) dV R D−5/2

(5.1.6)

where, the primed coordinates (x , y , z ) represent the location of source, n = 1− D2 , R is the distance between any point in the source to observation point, Γ (x) is the (2) gamma function and Hn (k R) denotes the Hankel function of second kind or order n representing outward going waves from source point. Now, to validate our provided solution in (5.1.6), we get vector potential F from our solution by substituting D = 3 in (5.1.6). For D = 3, we have n = −1/2. Using Hankel function of fractional order in (5.1.3), finally Eq. 5.1.6 gets reduced to ε e− jk R (5.1.7) FD = C dV M(x , y , z ) 4π R Equation 5.1.7 is in exact agreement with the solution provided in [1] for Euclidean space.

5.1.3 Radiated Electric and Magnetic Fields in Far Zone for Electric J and Magnetic Current Source M In the previous Sections, we have developed equations that can be used for electric and magnetic fields generated by and electric current source J and a magnetic current source M in fractional space. The procedure for radiation analysis in fractional space requires that the potential functions AD and FD are generated, respectively, by J and M. In turn the corresponding electric and magnetic fields are then determined in far zone. The fields radiated in far zone (k R >> 1) due to AD are EAD and HAD and are given by [1] EAD = − jωAD HAD =

− jω aˆ r × AD η

(5.1.8) (5.1.9)

where η is wave impedance. The fields radiated due to FD are EFD and HFD and are given in the same form as (5.1.8) and (5.1.9): HFD = − jωFD EFD =

− jω aˆ r × FD η

(5.1.10) (5.1.11)

64

5 Electromagnetic Radiations from Sources in Fractional Space

And finally, the total fields are given by superposition of the individual fields due to AD and FD as: ED = EAD + EFD

(5.1.12)

HD = HAD + HFD

(5.1.13)

5.2 Elementary Hertzian Dipole in Fractional Space Consider a Hertzian dipole ( an infinitesimal linear wire with l

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